_________________________________________________________________________________________________________

Electrical Engineering DepartmentLos Angeles, California 90095-1594

UNIVERSITY OF CALIFORNIA • LOS ANGELES

Role of Trivelpiece-Gould Waves in AntennaHelicon Wave Coupling

Donald ArnushUCLA

LTP-003 March, 2000

Low Temperature PlasmaTechnology Laboratory

1

The role of Trivelpiece-Gould Waves in Antenna Helicon Wave CouplingDonald Arnush

Electrical Engineering Dept., University of California, Los Angeles, California 90095-1594

ABSTRACT

It is well known that the simple theory of helicon waves, in which the electronmass me is neglected, is valid only if Ez also vanishes, a condition which is notsatisfied in experiment. Exact solutions of cold plasma theory with finite me and Ezpredict the existence of additional highly damped Trivelpiece-Gould (TG) modes,which can greatly modify the nature of helicon discharges. However, mostexperiments have been explained using only the simple theory for which the heliconwaves are undamped. In that case, antenna-plasma-coupling calculations predictinfinite resonances. To avoid this problem theorists have set Ez = 0 and includedfinite me effects (the TE-H theory). By comparing the TE-H theory with exact (i.e.,closed form) solutions for uniform density, the role of TG modes has been clarified.To do so for non-uniform density, a new algorithm is developed to treat the case ofhigh magnetic fields, when the wave equation becomes singular. The results showthat, though the wave patterns are not greatly affected by TG modes except at lowmagnetic fields or near the radial boundary, the kz-spectrum and radial profile of theenergy deposition are greatly modified. In particular, the peaks in the TE-H-modespectrum, which lead to predictions of erroneously high antenna loading, aresuppressed and broadened by the TG modes which also predict high edge absorption.Both the TE-H and exact theories give maximum antenna loading for ½me(ω / k)2 ≈10 - 100 eV, in contrast to several hundred eV predicted by the simple theory.

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I. INTRODUCTIONPlasma sources employing helicon wave excitation are of current interest because

they may provide the high density and uniformity needed for the fabrication of the nextgeneration of semiconductor circuits1,2. It has been known for some time3 that finite electronmass me, couples helicon to electron cyclotron Trivelpiece-Gould (TG) waves4,5,6 in uniformplasma. Despite that fact, most early experiments were interpreted using a simple model ofhelicon waves in which me, and hence δ = ω/ωc is neglected7,8,9,10,11. More recently Shamrai,et al12,13,14 have investigated antenna coupling to a uniform density plasma and explainedwhy, for most cases of interest, the wave magnetic fields, which are the easiest to measure,are dominated by the helicon waves. They also showed that the major channel for plasmaheating is coupling to helicon waves, which are weakly damped throughout the body of theplasma, and “mode conversion” from helicon to TG waves near the surface. The TG wavesare strongly damped. Kamenski and Borg15 treated antenna coupling to non-uniform plasmatheoretically, assuming that the axial electric field vanishes, and neglected TG waves. Arnushand Chen16,17 have more recently calculated antenna coupling to a non-uniform plasma at lowto moderate static magnetic field ( B0 1~< kG ) and density ( n0

13 310~< −cm ), to first order in δ,and found results in agreement with Shamrai et al for nearly uniform density profiles. Formore centrally peaked profiles (e.g., for a parabola) they found that the large radial TGcurrents near the surface are canceled by helicon currents and consequently the plasma centeris more intensely heated. In contrast to the prior studies of cylindrically bounded plasmas,Borg and Boswell18 considered wave excitation by an infinite line source in infinite uniformcollisionless plasma. They found that the TG mode does not lead to a significant increase inantenna coupling, and that the resonance cone is not a location of spatially resonant powerflux.

In this paper the antenna coupling to cylindrically symmetric plasma is consideredusing a collisional cold plasma dielectric. The role of TG wave damping is explicitlyexplored by comparing uniform density, exact solutions with solutions using a transverseelectric (TE) approximation that includes only helicon waves (TE-H), similar to thatinvestigated by Kamenski and Borg. For non-uniform densities and low to moderatemagnetic fields ( 0 300GB <! for a 5 cm radius plasma) the plasma field equations can besolved numerically by standard methods. At higher magnetic fields the equations aresingular (i.e., the coefficient of the highest order derivative tends to zero with increasing B0)and an approximate solution is obtained and investigated. The main focus of this paper is adiscussion of the dependence on density and magnetic field of the power spectrum of wavesabsorbed by the plasma, the radial distribution of the power absorbed, and the antennaresistance. These quantities are featured primarily because their understanding is key todesigning optimal helicon sources, and also because the presence of TG waves has arelatively small effect on the spatial dependence of the observed magnetic wave fieldsmaking it difficult to measure.

II. GENERAL FORMULATION

A. Boundary Conditions

Consider a cylindrical plasma of radius a, which is uniform axially and non-uniformradially, and which is enclosed in a conducting cylinder of radius c, with an intervening thinshell antenna of radius b, such that a < b < c. There are two linearly independent sets of

3

solutions that are finite on the axis. These plasma “basis functions”, are distinguished by theirbehavior near the axis and denoted by lower case symbols (e, b and j for E, B and J) and thesubscripts 1 and 2 (for a uniform plasma 1 denotes the helicon and 2 the TG wave). Take theFourier transform of each component Vi, of any wave quantity to be

2

( )

0

( , , ) ( , , )2

i m k zii

dV r m k dz V r z eπ

φφ φπ

∞− +

−∞

= ∫ ∫ . (2.1)

Variables are underlined to distinguish configuration-space from k-space functions. In Eq.(2.1) and the following we use zk k≡ . There are ten unknown basis function coefficients (2vacuum regions × 2 polarizations × 2 behaviors at 0 and infinity + 2 plasma basis functionsets). Using the usual ten boundary conditions ( tan 0=E at r = c, is continuous at r = b and r= a; tanB continuous at r = a but undergoes a jump at r = b) and the fact that in the vacuumspaces we have

2 2 2 20 0 0 0

and Bz z z zmk mkE E i B B i ErT T r rT T rφ φ

ω ω∂ ∂= + = −∂ ∂

, (2.2)

where 2 20 0T k k= − is the transverse wave number in the vacuum and k0 = ω/c, Arnush and

Chen17 have shown that Vi(r) can expressed as

2 1, 1 2,0

1 2 2 1

v ( ) v ( )( ) i i

i

H r H rV r i K

FG F G φµ−

=−

. (2.3)

The m and k functionality is understood. Kφ is derived from the antenna current density ofthe form J K( , , ) ( ) ( , )r z r b zφ δ φ= − (the surface current is assumed to be closed so that( / ) 0zm b K kKφ + = ) and

, , 0 0 ,20 0

20

, , 0 , ,20 00

20

0 20 0

( ) ( ) ( ) ( ) ,

1( ) ( ) ( ) ,

( ) ,

n n r m n z n z

n n r n z m n z n r

n m n m n

k mF b a p a b a e aT aT

km kG j a b a q e a ea TT

kkb mH p b G FT a a T

µ ωε

ωεµ

µ ρ

= + +

= + + −

= −

!

! ! (2.4)

where

(2.4a)

(2.4b)

(2.4c)

4

0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

( ) ( ) ( ) ( )( ) ,( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ,( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ,( ) ( ) ( ) ( )

m m m mm

m m m m

m m m mm

m m m m

m m m mm

m m m m

K T r I Tc K T c I T rp rK T a I T c K T c I T a

K T a I T c K T c I T aqK T a I T c K T c I T a

K T b I T c K T c I T bK T a I T c K T c I T a

ρ

′ ′ ′ ′−=′ ′−

′ ′−=−

−=−

(2.5)

Equations (2.4), (2.5) and the forthcoming text are somewhat simplified from Arnush andChen17 by the use of , and r r r rb ib b ib e ieφ φ= = =! ! ! . Note that the term ,n re! in the expressionfor Gn was inadvertently omitted in the original published version of this equation.

B. Wave EquationsUsing the Krook-modified dielectric tensor ΚΚΚΚ = εεεε/ε0, including electron-ion and

electron-neutral collisions and the notation of Stix19: ,xx yy Sε ε= = ,zz Pε = and

yx xy iDε ε= − = , the equations for the wave fields are

( )

0

0

2

0 2 20 0

2 200

0 0

0

0 0

1 ,

,

1 1 1 ,

1 ,

1 ,

.

r z

z r

z

z r z

r z

r z

mE E E k cBr r r

E kE k cBr

m k mB E B k P Er r k c r k r c

k m kB D E k k S E Er c k c r k c

mB E kEk c r

D kc m cE E B BS k S r k S

φ φ

φ

φ φ φ

φ

φ

φ φ

∂ = − +∂∂ = −∂

∂ = − + + − ∂ ∂ = − + − +∂

= −

= − + −

! !

! !

! !

! !

! ! !

(2.6)

The first four of Eqs. (2.6) form a closed set and can be solved numerically by standardmethods except at large magnetic fields. If the density is uniform the basis function fieldsare the familiar Bessel function solutions given by

( )( )

,

4 2 2 2 2 2 2 20 0 0

( ) ( ), where

( ) ( ) 0,

, ( ) / 2.

n z m n

n n

b r J T r

ST T k S P k SP RL P k R k k k L

R L S D

=

+ + − + − − − =

= ±

(2.7)

(2.7a)

(2.7b)

(2.7c)

(2.5a)

(2.5b)

(2.5c)

(2.6a)

(2.6b)

(2.6c)

(2.6d)

(2.6e)

(2.6f)

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III. SPECIFIC SOLUTIONSA. Transverse Electric Helicon Solution (TE-H)

If we assume that 0zE ≡ in the plasma, then only the TE mode exists in the cavity.Including the single, helicon, plasma mode there are five basis functions and hence fiveunknown coefficients. Three of the general boundary conditions are automatically satisfiedso that there are seven boundary conditions, and the system of equations for the coefficientsis over determined. We derive the TE-H analog of Eq. (2.3) without the use of the boundaryconditions on Bφ. The consequences of this action are discussed in section C. The resultsare:

0(0)

0

(0)

0

( )( ) ( ) ,

( ) ( ) ( ).r z

r kbr i p b KF T a

kF b a p a b aT

φµ

=

= +

vV

!(3.1)

A TE-H resonance as a function of k therefore occurs for F(0) = 0, in contrast to the simpletheory condition7 br(a) = 0.

If we set 0zE ≡ in Eqs. (2.6) The second equation becomes algebraic and the set isover determined. However, we note that in the limit me → 0 , P ∝ me

-1 and Ez ∝ me.Consequently, the product, and hence the equation is indeterminate. We therefore eliminatethe equation from the set and obtain the following:

( )220

0 2 22 2 200

220

32 20 0

02 2

0 0

,0

20

2 20 0

11 1 ,

1 ,

,

.

z

z z

z

r

r z

k DmE k cB m Er r k k Sr k k S

k DmB B T Er r k k S ck

kk D mB E Bk k S c k r

kB Eck

k mE DE cBk k S k r

φ φ

φ

φ φ

φ

φ

∂ = + + − ∂ −− ∂ = − −∂ −

= + −

= −

= + −

!

! !

! !

! !

(3.2)

In Eq. (3.2) we have used

( )( )2 2 2 20 02

3 2 20

.k R k k k L

Tk k S

− −=

− (3.3)

For a uniform plasma the solution to Eqs. (3.2) is 3( ) ( )z mb r J T r= . T3 can be obtained fromEq. (2.7) by taking the limit P → ∞ and self consistently neglecting T4. Note that thissolution differs from the 0em → limit since in that case S = 0, D is real, and hence T3 is real.Consequently, the resonances of Eq. (3.1) are infinite. The inclusion of these first orderterms provides dissipation which limits and broadens the resonance. In addition, T3 is a good

(3.2a)

(3.2b)

(3.2c)

(3.2d)

(3.2e)

(3.1a)

(3.1b)

6

approximation to the helicon root of Eq. (2.7), except at long wavelengths well above the cutoff for helicon wave propagation. For non-uniform plasma, the closed set of the first twoequations of (3.2) are solved as in the general case.

B. Helicon Trivelpiece Gould Solution (H-TG)If the density is uniform the closed form Bessel solutions are used. If the density is

non-uniform, and B0 is not too large (typically B0 < 300 G) the solutions of (2.7) are used inthe first radial step from the axis and Eq. (2.6) is integrated outward to the plasma surface. IfB0 is sufficiently large the equation set (2.6) becomes numerically difficult to solve and weresort to solving it using a series in me.

If we integrate Eq. (2.6) starting near the axis, each basis function will be a linearcombination of TG-like and helicon-like waves. At large magnetic fields the TG radial wavenumber can be approximated by 2 2

2 /T k P S≅ − . If 2Im 1T a " both solutions are dominatedby the TG-like functions, then Eq. (2.3) requires the calculation of small difference of largenumbers. To avoid the difficulties inherent in that process note that any two linearlyindependent solutions of Eqs. (2.6) may be used in Eq. (2.3). We therefore choose thesolutions to Eqs. (3.2) for the first, helicon-like basis functions. The second, TG-likesolution is obtained by an expansion in me. For me of order ε, S and P are of order ε and 1ε −

respectively in the limit 0ε → . We expand the field components in the following series

,0

exp ni i n

nV i Aψ ε

ε

∞

=

= ∑ . (3.4)

If we require a non-zero solution and calculate to order ε0 we obtain

( )

( )

1/ 42

2 20

2, 2, 2, 2,

2, 2, 2, 2,0

, ,

( )( , ) ,

2

( )0, exp ( , ) , ,( )

( , , ) ( , , ) .

a

r

r z z

r z z

P Dkr S PSr

m Dk PS Dr rk r i dr

k PS

r iSb b k r b ba kDc me e e i k b

k D r

φ

φ

ψψ β

ψ

ββ

ψ

∂ −′ = = =∂ −

∂ − + ′ ′∂′ ′Ψ = − + −

= = Ψ = −

′= − − − −

∫

! !

!!

(3.5)

C. Relationship of the solutionsNeglecting terms of order (k0/T0)2, as well as the displacement current terms (in

square brackets) in Eqs. (2.4), Eq. (2.3) becomes

2, 1 1, 20(0) (0)

2, 1 1, 2 0

( ) ( ) ( ) ( )( ) ( )

( ) ( )r r

r r

j a r j a r kbr i p b Kj a F j a F T a φµ

− = −

v vV , (3.6)

Which is valid provided that the outer cylinder-plasma gap, c-a, is not too small17.Resonance occurs as a function of k if the denominator of Eq. (3.6) vanishes. Anti-resonances occur if either one of the basis function radial currents vanishes20. Thus, if for a

(3.5a)

(3.5b)

(3.5c)

(3.5d)

7

given k, 1, ( ) 0rj a = (equivalent to the simple helicon boundary condition for uniformplasma), only helicon waves are present and we recover Eq. (3.1). However, except for rarecoincidences, the amplitude of the helicon wave is reduced since (0)

1F will not be near zero atthat k. For high magnetic fields ( / 1ceδ ω ω= << ), Im 2T ( 1 2 and T T are the helicon and TGradial wave numbers) is very large and the second term in the numerator of Eq. (3.6) isnegligible compared to the first except near the surface. Similarly, since 0 , ,i r i i rj Tbµ ≅ (therelationship is exact for a uniform density) the magnitude of the ratio of the second to firstterm in the denominator is of the order of 1 2/ 1T T << . Thus, neglecting the second terms inthe ratio, Eq. (3.1) is recovered from Eq. (3.6) in the helicon approximation. However, incalculating the configuration space fields from these equations by taking the Fouriertransform, the results are dominated by resonances due to the vanishing of the real partof (0)

1F . The imaginary part of this quantity, related to the imaginary part of 1T , is very smallsince the helicon wave is undamped to lowest order in em . On the other hand, the imaginarypart of (0)

2F is significant since the TG wave is heavily damped. Thus, to meet the boundaryconditions, both helicon and TG waves must be induced at the plasma surface. The TGwaves reduce and broaden the resonance and significantly modify the power coupling.

Consider now the consequences of ignoring the boundary conditions on the magneticfield in deriving Eq. (3.1). Using ABφ∆ as the value of Bφ just outside the antenna, minus thatjust inside, the jump condition across the antenna is satisfied if the following quantityvanishes:

20

0 020

Az

kB K KTφ φµ µ∆ − = (3.7)

Since, for most cases of interest, 20( / )k T is very small, the jump condition is satisfied in the

sense that the correction is much smaller than any of the individual terms in the equation.Defining the difference in Bφ at the plasma boundary as PBφ∆ in a similar fashion, we obtain

020

( )Pr

kB i j aTφ µ∆ = − . (3.8)

The simple theory was derived for uniform plasma where jr is proportional to br. The simpleboundary condition therefore is equivalent to setting PBφ∆ , and hence the radial current, equal

to zero. In the general theory we set 0PBφ∆ = and an axial current carried by the TG wavesbalances the radial current near the surface. In the TE-H theory, since the radial current isnot necessarily equal to zero it must be balanced by an ad hoc axial surface current giventhrough Ampere’s law by 0 ( )P

z zB K aµ∆ = .

IV. ANALYSISA. Measures of Power Coupling

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We define the Specific Plasma Power Spectral function, ( )PS k ′ as the powerabsorbed by the plasma for an antenna with ( )K k kφ δ ′= −

* 31( ) Re .2P plasmaS k d r′ = •∫E J (4.1)

Since ( )PS k ′ is an even function, for one ampere the Plasma Power Spectral function Pp(k),and Plasma Resistance Rp, is

( ) 2 2 20

0

02 2

( ) ( ) ( )

Im 2 Re ,2

2 ( ) where

( ) ( ) ( ) .

P P Aa

r z r

P P

A

P k S k p k

S E E P E D E E rdr

R P k dk

p k K k K k

φ φ

φ φ

ωε ∗

∞

=

= + + +

=

= + −

∫

∫

! !

(4.2)

There has been considerable speculation in the literature about helicon waves acceleratingelectrons with the velocity ω/k by the Landau damping mechanism. It is therefore useful toexpress Pp(k) as a function of the energy of the electrons u = ½me(ω/k)2 using a variabletransformation, as follows:

2 2

3

0

( ) ,8 2

2 ( )

e eu k

p u

m mP u Pu u

R P u du

ω ω

∞

=

= ∫(4.3)

Similar to Pp(k), the Antenna Power Spectral function PA(k), and Antenna LoadingResistance RA, can be defined as

* 3

0 0 0

0 0

0

1( ) Re2

( ) Im ( )2

2 ( ) .

A antenna

z zA m m

r a

A A

P k d r

c k a kB m Ep k p bT K ka cT K

R P k dk

φ φ

µ ρ=

∞

= − •

= +

=

∫

∫

E J

!(4.4)

Note that for most cases of interest the second term in the expression for PA(k) is negligibleand hence PA is proportional to the surface axial magnetic wave field. As we shall see, athigh magnetic fields the spectrum is dominated by 1 or a few resonances and the antennaresistance may be estimated from the sum of the residues at the resonances. If no powerescapes axially (due to either end reflections or total absorption in a very long cylinder) then

( ) ( ) and .A P A PP k P k R R= = (4.5)

(4.4a)

(4.4b)

(4.3a)

(4.3b)

(4.2a)

(4.2b)

(4.2c)

9

Indeed, the degree to which these equalities are met is a good measure of accuracy of basisfunctions either numerically integrated or approximated. It is also instructive to considerPr(r), the power absorbed in a differentially thin radial shell the length of the cylinder

( ) 2 2 20

0

( ) Im 2 Re ,2

2 ( )

r r z r

a

P r

P r S E E P E D E E dk

R P r rdr

φ φωε ∞

∗

−∞

= + + +

=

∫

∫

! !

(4.6)

To illustrate the results of this formulation the following parameters were selected forargon, using a collision frequency calculated from the electron-ion analytic form and theelectron-neutral cross-section21 convolved with a Maxwellian electron distribution:

0 e 0a=5cm, b=5.5cm, c=15cm, f 13.56MHz, T =3eV and P =3mTorr= . (4.7)

A half-turn helicon antenna is used, and results are quoted for the m=1 azimuthal Fouriercomponent. Bessel functions of a complex argument, ( )mJ z , are calculated using a powerseries for 11z < and an asymptotic series22 for 11z ≥ . For a radially non-uniform density,

for each k, Eq. (2.6) is integrated if Re ( ,0) 23kΨ < ( ) 10(i.e., exp ( ,0) 10 )kΨ < , and Eqs.

(3.2) and (3.5) are used if Re ( ,0) 23kΨ ≥ . It is necessary to select a sufficient number ofradial integration points N, to adequately sample the rapid radial oscillations of the TGwaves. It was found that the fractional error in the calculated resistance ( ) /A p AR R R−decreased at the approximate rate 1/N1.75. For all calculations N was chosen large enough toproduce an error less than 1%. The cylinder is assumed to be sufficiently long to sample thenarrowest spectral resonance. The calculation is performed using a C-based PC computerprogram, with a graphical interface, called HELIC. It implements the analysis describedherein for a wide range of parameterized density profiles and calculates the various powerfunctions and resistances, as well as the electric and magnetic fields and the current as afunction of z for fixed r, and a function of r for fixed z. Simple plots of these functions arealso provided. The code is currently being beta tested for later release.

B. Uniform Density

Consider first uniform densities for which closed form solutions are available. Thepower spectral functions and radial absorption profiles for the TE-H and H-TG solutions areshown in Fig. 1 for a plasma density n0 = 1011 cm-3 and B0 = 25 G. Even at this low densityand magnetic field the TE-H solution absorption profile is more sharply peaked than theanalytic one. The respective antenna resistances are 5.67 Ω and 1.39 Ω, a ratio of about 4.Apparently the TE-H approximation yields a larger loading resistance because of thespuriously high peaks in the k-spectrum. Both solutions predict peak absorption on axis withthe TE-H displaying a larger peak relative to the surface. Figure 2 shows the results for n0 =1012 cm-3 and B0 = 100 G. Both solutions have two dominant resonant modes atapproximately the same wave number. The TE-H solution displays an additional, lowerpower, long wavelength mode. The TE-H and H-TG antenna resistances are 2.97 Ω and 1.20Ω, a reduced ratio of about 2.5. The TE-H peaks have increased and narrowed, but theintegrated power has not increased correspondingly. The TE-H absorption still peaks on

(4.6a)

(4.6b)

10

axis, but has a flat plateau starting about midway to the surface. The H-TG profile peaks onthe surface and decreases exponentially radially inward, indicating the dominance of the TGwaves in absorption for these parameters. Approximately 75% of the power are deposited ina 1 cm thick surface layer. The pattern is continued for n0 = 1013 cm-3 and B0 = 1000 G,shown in Fig. 3, where the TE-H and H-TG resistances are 3.21 and 1.81 Ω. In that case thewidth of the TE-H spectral resonances are of the order of 0.2 m-1. In an ideal cavity, wherethe axial cavity modes are separated by π/L (L is the cavity length), the cavity would have tobe 15 m long to insure that the TE-H modes are excited. Thus, a major result of thebroadening of the resonances by the TG modification of the denominators in Eqs. (2.3) and(3.6) is to permit excitation of the helicon modes in cavities of acceptable length. For plasmastability 0/ 0R n∂ ∂ ≤ at constant B0 is required23. The peak of R(n0,B0) for B0 = 1000 Goccurs for n0 = 4 ×1012 cm-3. The H-TG power absorption as a function of acceleratedelectron energy is shown in Fig. 4 for n0 = 4, 6, 8, 10 ×1012 cm-3. The curves all diminishrapidly for u > 100 eV even for the cases where long wavelength secondary peaks arepresent, due the factor u-3/2 in Eq. (4.3). The corresponding result for the TE-H solutionshows large peaks with widths less than about 1 eV. Thus, the H-TG theory predicts that thewaves are preferentially absorbed in a broad range of energies from about the ionizationthreshold up to about 100 eV, depending on the density. This contrasts with the TE-H theorythat suggests the presence of electron beams with narrow energy spreads. Calculation of theradial dependence of the wave magnetic field 20 cm from the antenna, B(r,0.2), for n0 = 1012

cm-3 and B0 = 100 G yielded only small deviations between the H-TG and TE-H solutions.They were located in the 1-cm surface layer. The effect of the broadening on the variation ofBz(0,z) for the same parameters is shown in Fig. 5. The magnitude of Bz calculated using theTG-H approximation clearly displays a beat phenomenon between the modes and a gradualdecay, as seen in experiments24. The calculated beating is less pronounced, and the decay ismore rapid with the H-TG formulation, due to interference between the wide range of k’s inthe broadened spectrum shown in Fig. 2. However, as seen in Fig. 5b, the phase issurprisingly little affected. Since there are many effects not included which can affect therate of spatial decay, it would be difficult to use the measurement of Bz(0,z) to distinguishbetween the H-TG and TE-H result.

Contour plots of the antenna resistances for the two approaches in the range 1012 < n0< 1013 cm-3 and 100 < B0 < 2000 G are shown in Fig. 6. The contours of constant R in theTE-H approximation are straight lines (i.e., 0 0B n∝ ). The ridge of maximum R (≈5.9Ω) isgiven approximately by B0(G) = 250 ×10-12 n0(cm-3). The local peak resistance of the H-TGcontour plot near n0 = 2.3 ×1012 cm-3 and B0 = 550 G is approximately 3.5 Ohm. Theresistance varies slowly along the ridge, dropping to 2.8 Ohm at the saddle point n0 = 6.5 ×1012 cm-3 and B0 = 1500 G, and rising to 3.6 Ohm at n0 = 6 × 1012 cm-3 and B0 = 2000 G.The ratio of resistance calculated using the TE-H and H-TG treatments is in the general range1.5 to 2. The ridge in the H-TG contour plot follows approximately a straight line for B0 <1000 G. The line intersects n0 = 8 × 1012 cm-3 at 1500 G. The spectra for B0 = 1500 G andn0 = 5, 6.5 and 8 × 1012 cm-3 are shown in Fig. 7. The resistances vary slowly (2.7, 2.8 and2.7 Ohm respectively) although there is significant variation in the spectral shapes.

C. Non-uniform Density

It is of interest to explore the effect of the radial density profile on the powerabsorption. Linear (1-r/w) and parabolic (1-(r/w)2) profiles, where w is adjusted to keep the

11

ratio of central to surface density equal 10, were calculated and the results compared to auniform profile. In the three cases shown in Fig. 8, B0 is 1000 G and n0 = 1013 cm-3 on axis.As the density profile becomes more centrally peaked, the resonances shift to smaller wavenumber and become narrower, but remain much broader than those predicted by the TE-Happroximation. The antenna resistance for the uniform, parabolic, and triangular profilesequal 1.81, 3.17 and 3.36 Ohm respectively. Much greater power dissipation on axis occursfor the centrally peaked profiles. This result is consistent with the experimental observationsthat density profiles for high power, high-density discharges are centrally peaked25 andapproximately triangular for the highest density cases. The high dissipation on axis sustainsthe high density there, and the high dissipation near the surface sustains the high power lossto the walls. Calculations were performed using the same profiles but with the centraldensities increased by a factor of 2 and 3 for the parabolic and triangular shapes so that totalamount of plasma was approximately the same for the three cases. The results were similarto those shown in Fig. 8, except that the antenna resistance only increased to about 2.5 Ohmfor the centrally peaked cases. The diminished enhancement is due to the fact that, atconstant magnetic field, the resistance decreases with density in this parameter range.

V. SUMMARY AND CONCLUSIONS

The H-TG solution of the uniform-density, collisional plasma model of antennacoupling was compared to a well-defined, transverse-electric, pure helicon model. Formoderate to large static magnetic fields ( 0 100B >! ) differences between the configuration-space wave magnetic fields predicted by the two solutions were found to be small and/ordifficult to measure. Differences in the spectrum and the radial profile of the powerabsorption, as well as the topography of the antenna resistance R(n0,B0), critical to optimalplasma source design, were found to be significant. The predicted TE-H resistance istypically somewhat less than twice the resistance predicted by the H-TG method. Thereduced resistance in the H-TG calculation occurs partly because the presence of TG waveschanges the predicted electric fields and currents, but more importantly because they modifythe charge and current balance at the plasma surface and consequently the height and widthof the plasma resonance. At large magnetic fields the predicted TE-H resonance is extremelynarrow, requiring a very long cavity to insure overlap between the resonance and the axialcavity modes at equilibrium, as well as in the temporal approach to equilibrium. Landaudamping for the H-TG solution is predicted to occur in a broad range of energies from about10 eV to about 100 eV, in contrast to the TE-H solution that predicts the presence ofnarrowly energetic beams. For a uniform density and large magnetic field, both modelspredict a small axial peak in the power absorption profile, with the H-TG solution predictingan additional sharp increase in the absorption in a thin surface layer. For more realistic,centrally peaked, density profiles the H-TG solution predicts a thickening of the surface layeras well as a much more pronounced peak in the heating on the axis.

ACKNOWLEDGEMENTS

The author would like to thank F. F. Chen for many helpful discussions and C-C. Lanfor writing the HELIC computer code.

12

Fig.1. H-TG vs. TE-H power absorption spectrum (a), and radial profile (b), for a uniformdensity n0 = 1011 cm-3 and magnetic field B0 = 25 G

a

P A(k

) (Ω

-m)

k (m-1)

H-TGTE-H

b

P r(r) (

Ω-m

-2)

r (m)

13

Fig.2. H-TG vs. TE-H power absorption spectrum (a), and radial profile (b), for a uniformdensity n0 = 1012 cm-3 and magnetic field B0 = 100 G.

k (m-1)

P A(k

) (Ω

-m)

a

0.001

H-TGTE-H

b

P r(r) (

Ω-m

-2)

r (m)

14

Fig.3. H-TG vs. TE-H power absorption spectrum (a), and radial profile (b), for a uniformdensity n0 = 1013 cm-3 and magnetic field B0 = 1000 G.

a

P A(k

) (Ω

-m)

k (m-1)

H-TGTE-H

b

P r(r) (

Ω-m

-2)

r (m)

15

Fig. 4 H-TG power absorption spectrum as a function of accelerated electron energy u =½me(ω/k)2 for n0 = 4, 6, 8, 10 × 1012 cm-3.

u (eV)

P u (u)

( Ω-e

V-1

)

4 × 1012 cm-3

6 × 1012 cm-3

8 × 1012 cm-3

1 × 1013 cm-3

16

Fig.5. H-TG vs. TE-H amplitude (a), and phase (b) of the axial wave magnetic field on axisas a function of z, Bz(0,z), for a uniform density n0 = 1012 cm-3 and magnetic field B0 = 100 G.

|Bz(0

,z)|

(T)

z (m)

aH-TGTE-H

arg

(Bz(

0,z)

(rad

)

z (m)

b

17

Fig.6. Contours of constant H-TG (a), and TE-H (b) antenna resistance (Ohm), as a functionof density and magnetic field.

n0 (cm-3)

B0 (

G)

a

n0 (cm-3)

B0 (

G)

b

18

Fig.7. H-TG power absorption spectrum at Bo=1500G and uniform densities n0 = 5, 6.5 and8×1012 cm-3.

P A(k

) (Ω

-m)

k (m-1)

12 3

12 3

12 3

5.0 106.5 108.0 10

cmcmcm

−

−

−

×

×

×

19

Fig.8. H-TG power absorption spectrum (a) and radial profile (b) for uniform, parabolic andtriangular density profiles. The magnetic field is 1000 G and the density is 1013 cm-3 on theaxis for each case.

UniformParabolaTriangle

P A(k

) (Ω

-m)

k (m-1)

a

P r(r) (

Ω-m

-2)

r (m)

b

20

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