Microwave plasma in non-uniform quasi-static fields with radial symmetry

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INSTITUTE OF PHYSICS PUBLISHING PLASMA SOURCES SCIENCE AND TECHNOLOGY

Plasma Sources Sci. Technol. 11 (2002) 146–151 PII: S0963-0252(02)34670-X

Microwave plasma in non-uniformquasi-static fields with radial symmetryYu A Lebedev, A V Tatarinov and I L Epstein

AV Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences,29 Leninsky Avenue, 117071 Moscow, Russia

E-mail: lebedev@ips.ac.ru

Received 5 July 2001, in final form 5 February 2002Published 5 April 2002Online at stacks.iop.org/PSST/11/146

AbstractThe characteristics of hydrogen plasma in quasi-static microwave fields withradial symmetry have been simulated. The balance equation for the chargedparticles, the equation for microwave electric field strength and theBoltzman equation have been solved self-consistently, and the stationarydistributions of plasma density and electric field settled as a result ofprocesses of ionization, diffusion and bulk electron–ion recombination havebeen obtained. The simulations have been carried out for hydrogen withparameter νen/ω within the range 0.15–1.73 and the power absorbed inplasma changed from 0.5 to 20 W. It was shown that for low absorbedpower, the discharge was presented by a thin electrode plasma layer with theelectron density everywhere below the critical value. At higher levels of theabsorbed power, the electron density was just above the critical value. Thesize of the plasma grew proportionally to further rise of the power, but theelectron density stayed at the same level. Simulations of the dissociation ofhydrogen have also been carried out. The dissociation was mainlydetermined by the fields inside the plasma formation, and not by a thin peakof the electric field near the internal electrode.

1. Introduction

The study of plasma which exists in strongly non-uniformmicrowave fields is one of the important problems in modernplasma physics. Such plasma can be generated for instancein initiated and electrode microwave discharges. The initiateddischarge [1, 2] is located around the initiator (usually a metalrod of different shape placed inside the electromagnetic field).In the electrode discharge [3–7], the electromagnetic energyenters the chamber along the electrode antenna. The dischargeis sustained at the edge of the antenna.

The structure of the electrode RF discharge has beentheoretically analysed [8, 9] for the case where the processeswhich define the radial distribution of the electron density weredirect electron impact ionization, diffusion and attachment.The appropriate formulae for the estimate of the thicknessof a plasma layer near the electrode have been derived. Thestability of different forms of the RF discharge in quasi-staticfields has also been investigated.

In [10] the dispersion equation for the wave propagationinside the medium partly filled with plasma of a given density

has been used for the description of the plasma column ina coaxial discharge system (dual-plasmaline). A numericalanalysis for the stationary discharge in such a system has beencarried out, with Eτ � En everywhere inside the plasma,where En and Eτ are normal and tangential field components.

In this paper, we present the results of numerical modellingof the hydrogen plasma in quasi-static microwave fields (Eτ =0) in systems with coaxial and spherical symmetry. It wasassumed that stationary distributions of the plasma density andfield settled as a result of processes of ionization, diffusionand bulk electron–ion recombination. In particular, theseconditions have been realized in experiments with the electrodemicrowave discharge [5]. To study the process of hydrogendecomposition, the radial distributions of hydrogen moleculesand atoms were also calculated. The work aimed at finding themain factors which defined the structure of the discharge.

2. Model

The system is presented by both coaxial and sphericaldischarge chambers (external electrodes) with cylindrical and

0963-0252/02/020146+06$30.00 © 2002 IOP Publishing Ltd Printed in the UK 146

Microwave plasma in non-uniform quasi-static fields

spherical internal electrodes, respectively. The radii of theexternal and internal electrodes are R and rel. The stationaryspatial distribution of microwave electric fields has settledinside the chamber partly filled with plasma due to the processof propagation and absorption of the microwave signal. In casethe electric field vector is always perpendicular to the gradientof plasma density, the amplitude of such a quasi-static electricfield in plasma with permittivity ε is given by [8, 9]

E(r) = E0

( rel

r

)k{(1 − n)2 +

ν2en

ω2· n2

}−1/2

, (1)

where k = 1 for a cylinder and k = 2 for a sphere; n =ne/nc is the electron density normalized to its critical valuenc = (

ν2en + ω2

)m/4πe2;ω is an angular frequency of the

microwave field; νen = νen(E) is a collision frequency ofelectrons and molecules; E0 is an amplitude of microwaveelectric field at the internal electrode of radius rel.

The electron balance equation with zero boundaryconditions on both electrodes is

1

rk

d

dr

(Dar

k d

drn

)+ νin − αrn

2 = 0, (2)

where Da is the ambipolar diffusion coefficient, νi is the directelectron impact ionization frequency, and αr is a coefficient ofthe bulk electron–ion dissociative recombination (where H+

3 isthe main ion [11]). The balance equation (2) with the boundaryconditionsn|rel = n|R = 0 describes the stationary distributionof plasma density within a given electrode system. The gastemperature was taken constant and equal to 300 K.

At each point r the quasi-static electric field E(r) wasdetermined and the stationary Boltzmann equation was solvedto find the electron energy distribution function (EEDF) andits moments. Then the coefficients Da, νi, αr and νen werecalculated.

The Boltzmann equation for the stationary, isotropicpart of the electron distribution, obtained in the two-termapproximation to the expansion of the distribution in sphericalharmonics, was used in the following form:

d

dε

[E0

3eε2

∑i (Niσ

itr (ε))

2eε∑

i (Niσitr(ε))

2 + mω2

df

dε

]= J el+J in+Jee+Jr,

(3)where J el, J in, and Jee are the integrals of the elastic, inelasticand electron–electron collisions, and Jr is the integral ofcollisions corresponding to excitement of rotational degreesof freedom of the molecules:

J el = d

dε

[2m

M

∑i

(Niσir (ε))ε

2

[f +

kTg

e

df

dε

]],

Jr = d

dε

[∑i

6εBiNiσitr(ε)

[f +

kTg

e

df

dε

]],

J in =∑i

∑α

Niα[εσ i

α(ε)f (ε)

− (ε + uiα)σ

iα(ε + ui

α)f (ε + uiα)]

+∑i

∑α

N∗iα [εσ i

−α(ε)f (ε)

− (ε − uiα)σ

i−α(ε − ui

α)f (ε − uiα)],

σ i−α = ε + ui

α

εσ iα(ε + ui

α),

Jee = 2υeeε3/2 d

dε

(f (ε)

ε∫0

ε1/2f (ε) dε +2

3

df

dε

[ ε∫0

ε3/2f (ε)dε + ε3/2

∞∫ε

f (ε) dε

]),

υee = 4πe4

m2ne

ln �

v3, ln � = ln

k2TeT1/2

g

e3n1/2e

.

Here, f (ε) is EEDF, ε is the electron energy, σtr, σα and σ−α

are the electron momentum transfer collision cross-section,the inelastic and superelastic cross-sections for collision of theelectrons with heavy particles, ui

α is the threshold of the α-thinelastic process for the ith heavy component, and Bj is arotational constant.

The method for solution of the Boltzmann equation usedby us is described in [16]. The set of cross-sections forcollisions of electrons with H2 particles (including directexcitation of the b3!+

u , a3!+g , C3"u, B1!+

u , E1!+g , c1"u, e3!+

u ,B1!+

u and d3"u electron levels, direct ionization, excitation ofthe two first vibrational levels, and the electron momentumtransfer collision cross-section and rotational cross-section)were taken from [17]. The processes of direct ionization anddirect excitation of the H(2S, 2P) state for atomic hydrogenwere taken into account [18].

The equation for the microwave power absorbed in plasmacompletes the system ∫

V

σE2 dV = P. (4)

Here σ = νenn/4π is the electron plasma conductivity. Theintegral is taken over the volume of the chamber.

The system of equations (1)–(4) has been self-consistentlysolved in the following order. First, in equation (1) theamplitude of the electric field on the surface of the internalelectrode E0 was chosen. Then equation (2) was solved bythe Runge–Kutta method at each point calculating coefficientswith the help of equation (3). Next condition (4) for given P

was checked. By varying E0 and repeating the previous steps,it was possible to get the system to converge.

On the whole, for given gas pressure, power absorbed inplasma and geometry of the electrodes, the electron density andmicrowave electric field radial distributions were found. In thecase of the cylindrical system of electrodes, we assumed thatn and E were not functions of z and that the length of plasmain the z-direction was 1 cm. The angular frequency of themicrowave field: ωw/2π = 2.45 GHz, and the parameter νenω

was within the range 0.15–1.73, which for such frequenciescorresponds to the hydrogen pressure: 0.5–8 Torr. The powerabsorbed in the plasma changed from 0.5 to 20 W and thenormalized radius of the external electrode changed from 10to 80. The simulations were mainly done for the externalelectrode of radius R = r/rel = 13.3, which was normalizedto the radius of the internal electrode: rel = 0.3 cm.

For known radial distributions of electron density ne(r)

and electric field E(r), it is possible to obtain stationary

147

Yu A Lebedev et al

radial distributions of concentrations of hydrogen atoms andmolecules from the following equations:

1

rk

d

dr

(DH2r

k d

drnH2

)− kdisnenH2 + krecn

2HN = 0, (5)

1

rk

d

dr

(DHr

k d

drnH

)+ 2kdisnenH2 − 2krecn

2HN = 0, (6)

where DH and DH2 are the diffusion coefficients for H andH2, N is the total number of heavy particles, kdis is thecoefficient of hydrogen dissociation under electron impact,and krec = 3 × 10−32 cm6 s−1 is the coefficient of bulkrecombination of hydrogen atoms. The values of kdis werecalculated from the Boltzmann equation in accordance withknown E(r). The simulations were done for p = constand N = const. In general, the total number of heavyparticles in a given volume element may change due to theprocess of dissociation. Such an effect would produce pressuredifferences in the gas. In simulations this problem is usuallysettled by renormalizing the concentrations of every species:the concentrations of H and H2 at each point r were normalizedto their sum.

System (5), (6) was solved with the boundary conditions:DHdnH/dr = γ υthnH/4 and DH2 dnH2/dr = −γ υth nH8,where υth is the mean velocity of heavy particles and γ isthe probability of surface recombination of hydrogen atoms(simulations were made for the stainless steel surface, γ =5 × 10−2 [12]) .

3. Results and discussion

The radial distributions of the electron density obtained fromsimulations reveal several features common for the wholerange of parameters νen/ω and power absorbed in plasma.There is only one peak of the electron density, which stays2–3 mm apart from the rod-shaped electrode. The densitymonotonously declines at both sides of the peak. The solutionsare automodelled at n < 1 (figures 1–3(a)).

In the case of low densities, i.e. n(r) � 1, the electrondensity grows linearly with the power absorbed in the plasmaat any νen/ω. The radial profile of the electric field declinesas 1/rk and does not change with rise of the absorbed powersignificantly. Once the peak of electron density exceeds thecritical value, we have to separate the cases when parameterνen/ω is less or higher than 1.

When νen/ω < 1 the maximum value of the electrondensity peak does not grow much with increase of power,but the supercritical area of the peak (n > 1) expands alongthe radius proportionally to the power (figure 1(a), curvescorresponding to 10 and 20 W). Inside this area, the densityslowly decreases from its maximum value towards the wall ofthe chamber. Outside the supercritical area, the density fallsmuch faster. The electric field has two maxima at n = 1. Thepeak value of the electric field is roughly ω/νen higher than thefield at the surface of the internal electrode. Such a structurecomes directly from formula (1). Inside the supercriticalarea, the electric field is almost constant (except at very closedistances from points n = 1) and it rapidly decays outsideit (figure 1(b), curves corresponding to 10 and 20 W). Theaveraged volume of the chamber electron density grows with

(a)

(b)

(c)

(d)

Figure 1. Radial distributions of (a) the electron density (nc is acritical density), (b) the electric field and (c) the absorbed power perunit volume inside the cylindrical system of electrodes for the totalpower P absorbed in the hydrogen plasma: P = 0.5, 1, 2, 5, 10 and20 W. Gas temperature T = 300 K, and νen/ω = 0.15. Circles:radial distribution of the intensity of emission from plasma. Theexperimental data are taken for the case P = 13 W, T ∼ 700 K andp = 1 Torr, which correspond to the regime of simulations. Plot(d) shows the radial distribution of the ionization frequency andmean electron energy for the case P = 20 W, T = 300 K andνen/ω = 0.15.

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Microwave plasma in non-uniform quasi-static fields

power due to the increase of the supercritical area of the plasma.Thus, it is impossible to achieve densities essentially higherthan the critical value by raising the input (and hence theabsorbed) power. Such behaviour seems to be general forunbounded plasma with n > 1 sustained by microwaves whenplasma is not limited in one or more directions by a dischargevessel. As an example, we refer to well-known phenomena inthe surface wave sustained plasma in the long tubes where thelength of the plasma slab increases with the input power.

In the case when νen/ω > 1, the peaks of electric fieldrelated to the plasma resonance are much less pronounced.The radial distributions of the electric field almost coincide inthe whole range of the absorbed power (figure 3(b)).

Maximum values of the electron density are smaller thanthose at low parameterνen/ω for the same levels of the absorbedpower and do not achieve the critical value. The electrondensity is linearly proportional to the absorbed power. Theprofile of the electron density has only one peak that is placedat a distance of 2–3 mm from the internal electrode. The widthof this peak is the same size no matter what the absorbed poweris (figure 3(a)).

It is known that the local light intensity of the emissionfrom plasma is related to the power density. We comparedthe radial distributions of the absorbed power (figure 1(c))taken from the simulations with the experimental datafor the electrode microwave discharge obtained by opticalmeasurements of the integral spectra emission with spaceresolution [5]. According to these measurements, the plasmaformation consisted of a thin bright layer around the internalelectrode and a spherical glow region. The latter appeared ata certain level of the input power in the whole range of thegas pressure. The size of the glow region grew with power.In simulations, the expansion of the plasma formation withthe absorbed power could be observed only in cases of highparameter νen/ω (low pressure) when the plasma resonancewas pronounced (figure 1,2(b, c), curves corresponding to10 and 20 W). These curves were similar to those observedin plasma emission experiments [5]. We put one of thesecurves corresponding to 13 W of absorbed power in figure 1(c)for comparison. It is also seen that from a certain levelof absorbed power, the main part of the absorbed power isconcentrated inside a sharp peak close to the internal electrode(50–80%). This is also in agreement with the results of opticalmeasurements. Appropriate parameters characterizing theelectron energy distribution function are shown in figure 1(d).For low parameterνen/ω (high pressure), the model did not givea similar structure though in the experiments the discharges ofsimilar shape and spatial structure were also observed. Thusthe model gives a qualitative description of the experimentaldata only in the case νen/ω > 1.

A few comments should be made on the plasma densityprofiles. In the experiments the density drops sharply (of anorder of magnitude) at the edge of the plasma formation [6].In simulations we made a few attempts to achieve this result.Instead of formula (2) the microwave electric field waspresented by a step-function with a step placed at the edgeof the plasma. However, we still had a smooth decrease ofdensity due to diffusion. It seems reasonable to somehowintroduce a direct field trap at the edge of the plasma to stopthe electrons from leaving the plasma formation and, hence,

(a)

(b)

(c)

Figure 2. Radial distributions of (a) the electron density (nc is acritical density) (b) the electric field and (c) the absorbed power perunit volume inside the spherical system of electrodes for the totalpower P absorbed in the hydrogen plasma: P = 0.5, 1, 2, 5 and10 W. Gas temperature T = 300 K, and νen/ω = 0.15.

produce a necessary density drop, as was assumed in [6].This field usually locates inside the double layer (DL), whichseparates the internal active plasma and external region of lowplasma density. The DL limits the plasma thermal expansionby means of the Maxwell stress in the electrostatic field [14],arising from the difference of particle moments per unit area persecond. Such an approach has recently been used in modellingthe structure of auroral phenomena in atmospheric plasma [15].The thickness of the DL is usually several mean free paths.There is a strong violation of the quasi-neutrality within such alayer and the appropriate electric field is quite large. A properdescription of the DL needs a transition to a model, whichworks in the absence of quasi-neutrality.

Simulations of the dissociation of hydrogen have beencarried out at low parameter νen/ω in the range of the absorbedpower 1–20 W. The maximum of the degree of dissociationη(η = 1 − nH2/N ) did not exceed 25%. The degree ofdissociation of hydrogen rises with increase of absorbed poweror decrease of pressure (figure 4). Sensitivity of simulations to

149

Yu A Lebedev et al

(a)

(b)

Figure 3. Radial distributions of (a) the electron density (nc is acritical density) and (b) the electric field inside the cylindricalsystem of electrodes for the total power P absorbed in the hydrogenplasma: P = 0.5, 1, 2, 5 and 10 W. νen/ω = 1.73 and T = 300 K.

Figure 4. Radial distribution of the relative concentration of H2

inside the cylindrical system of electrodes for γ = 5 × 10−2: (1)νen/ω = 0.3, P = 10 W; (2) νen/ω = 0.15, P = 10 W; (3)νen/ω = 0.15, P = 20 W. For comparison, curves (4) and (5)corresponding to γ = 2.5 × 10−2 and γ = 7.5 × 10−2 werecalculated for the same conditions as (3). T = 300 K.

the absolute value of the probability of surface recombinationof hydrogen atoms γ is illustrated in figure 4 (curves 4 and5). The radial profiles of H2 have a specific bell-like shapewith one minimum, which is shifted from the maximum of theelectric field due to diffusion.

The simulations showed that the contribution of theprocess of bulk recombination is not essential. Enlarging

(a)

(b)

Figure 5. Radial distributions of (a) relative concentration of H2

and (b) the electric field inside the cylindrical system of electrodesfor νen/ω = 0.15, T = 300 K, γ = 5 × 10−2. Curves (1) in (a) and(b) correspond to the case where the electric field is calculated fromformula (2) (as in all previous simulations). Curves (2)–(4)correspond to E given by a step-function. The heights of thestep-functions are chosen in such a way that the correspondingabsorbed power of cases (1)–(4) is of the same value P = 20 W.

the size of the chamber does not change the minimum of thedissociation coefficient.

A peak of the electric field situated near the internalelectrode (figure 5(b), curve 1) weakly affects the maximumvalue of the degree of dissociation because of the high rateof diffusion losses of heavy particles in the vicinity of theelectrode. The dissociation is mainly determined by fieldsoutside the nearby layer, which is clear from the comparisonof radial profiles of the dissociation of hydrogen for differentcases: with the electric field calculated from formula (2)(figure 5(b), curve 1) and the others with E given by a modelstep-function of different length (figure 5(b), curves 2–4). Theheight of the step-functions was calculated in such a way thatthe absorbed power was the same as that for the electric fieldcorresponding to curve 1 (figure 5(b)). The difference in thedegree of dissociation is not significant for all cases.

Spatial structures of the electric field and electron densitiesin the coaxial system of electrodes are similar to those inthe spherical system, except that in the last case they areconcentrated closer to the internal electrode (figure 2(a), (b)).In the vicinity of the internal electrode, the electric fieldand electron density are higher than those in the cylindricalproblem; however, they decay more rapidly towards theperiphery. This behaviour is mainly due to 1/r2 dependence

150

Microwave plasma in non-uniform quasi-static fields

of the electric field (see (1)) in the spherical problem. Theregion of main energy consumption is twice larger.

To reveal the influence of different parameters on thesolution of the problem, additional simulations were carriedout. It was shown that the coefficients D(E), αr(E) andνen(E), if they were everywhere taken as constants, do notsignificantly change the spatial distributions except at closedistances of the wall of the chamber, where E < 100 V cm−1.The difference in values was 10–20%.

The results of simulations are insensitive to the changeof the radius of the chamber. The decrease of the radius ofthe internal electrode leads to increase of the peak value ofthe electric field. This result agrees with the experimentalevidence of the fact that the discharge sustained at a thininternal electrode exists at a lower level of microwave powerthan that at the thick electrode [5].

4. Summary

The spatial distributions of the electron density in the hydrogenplasma in quasi-static microwave fields inside differentelectrode systems with radial symmetry were calculated. Themodel included the balance equation for the charged particles,the equation for microwave electric field strength and theBoltzmann equation, which were solved self-consistently. Insimulations for hydrogen, the parameter νen/ω within variedthe range 0.15–1.73 and the power absorbed in plasma changedfrom 0.5 to 20 W. For low absorbed power, the discharge ispresented by a thin electrode plasma layer with the electrondensity everywhere below the critical value. As the level ofthe absorbed power is high enough for the electron density toreach the critical value, in cases when the plasma resonancecan exist, the structure of the discharge changes. Further riseof the power leads to the radial expansion of the dischargewith electron density just above the critical value. In this casethe model has revealed a qualitative agreement of the radialdistribution of absorbed power with the experimental data,obtained from optical measurements of the integral spectraemission from plasma. The proposed model gives a reasonabledescription of the plasma in the range of parameters, whichcorrespond to the plasma resonance: νen/ω < 1, Pabs > 10 W.For commonly used microwave frequency f = 2.45 GHz, thegas pressure of hydrogen is less than 2 Torr and the electrondensity is higher than critical: 7.0 × 1010 cm−3. Qualitatively,the results for cylindrical and spherical systems of electrodesare not considerably different. In the spherical system, all

spatial distributions are more driven towards the internalelectrodes. For better agreement of plasma density profileswith those from experiments, the concept of a DL must beintroduced in the model and needs further thorough research.

Simulations of the dissociation of hydrogen have beencarried out at low parameter νen/ω in the range of the absorbedpower 1–20 W. The maximum of the degree of dissociationdid not exceed 25%. It was shown that the dissociation wasmainly determined by the fields inside plasma formation andnot by a thin peak of the electric field situated near the internalelectrode.

Acknowledgment

This study was partly supported by NWO Grant 047.011.000.01.

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