What are helicons?

Helicons are partially ionized RF discharges in a magnetic field.

They are basically whistler modes confined to a cylinder.

They are much different than in free space; they have E-fields.

OLD

NEW

Long cylinder Permanent magnet

Helicons pose unending problems

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• Why does the amplitude oscillate along the cylinder?

• Why is a right-helical antenna better than a left one?

• What causes the high ionization efficiency?

• Why does an endplate near the antenna increase n?

• Why is the ion temperature so high?

• Why is a half-wavelength antenna better than a full?

• Why is the density peaked at the center?

Most discharge theorists treat only collision cross sections and ion distribution functions.

The Trivelpiece-Gould mode: edge ionization

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Trivelpiece-Gould mode

Helicon mode

An electron cyclotron wave near the edge deposits most of the RF energy

0

1000

2000

3000

4000

5000

0.000 0.005 0.010 0.015 0.020 0.025r (m)

P(r

)

(arb

.)

2.5E+11

4.0E+11

6.3E+11

1.0E+12

n (cm-3)

Edge ionization should give a hollow profile

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B

n

r

0

1

2

3

4

5

-25 -20 -15 -10 -5 0 5 10 15 20 25r (cm)

n (1

01

1/c

m3),

KT

e (

eV)

0

2

4

6

8

10

12

14

16

18

Vs (V

)

n11KTeVsVs(Maxw)

65 Gauss

But density is almost always peaked at center, even in KTe

is peaked at the edge.

Previous attempt for an ICP

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Skin depth

with FL

without FL

Let’s take the simplest realistic problem

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Eliminate all unnecessary features, and not length!

L

aB

+

rLi >> a

rLe << a-

Treat a 1D problem in radius r

The problem is how to treat the ends

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HIGH DENSITY

LOWER DENSITY

SHEATH

B

+

e

e

+

½ ½ 1/2

, ln2 2

pe pe e

e

eKT KT Mn n eM m KT m

The sheath drop is normally independent of density

Ion diffusion upsets the balance

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HIGH DENSITY

LOWER DENSITY

SHEATH

B

+

+e

e

APPARENT ELECTRON FLOW ION DIFFUSION 1

2

(a)

The short-circuit effect “moves” electrons across B.

Sheaths change to preserve neutrality.

Electrons can now follow the Boltzmann relation.

This happens in nanoseconds.

Sheath drops interchange, creating Er

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HIGH DENSITY

LOWER DENSITY

SHEATH

B

+

+

e

e

ION DIFFUSION+

-

E

(b)

1

2

/0

ee KTn n e

( 0)

In equilibrium, n is peaked on center

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BSHEATH

+

LOWER DENSITY 2

e

+

HIGHDENSITY 1

e

+ e LOWEST DENSITY 3

ION DIFFUSION

+

-

E

Er and diffusion must be outward if axial flow is slow.n(r) is flat in the limit of all ionization at edge.

Three equations in 3 unknowns: v, n, and

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( ) ( ) /c cx io nP r v r n

( ) ( ) 0io iM n Mn en Mn en KT n v v v Bv v E v

Ion equation of motion:

Ion equation of continuity: ( ) ( )n in nn P r v

( ) ( )i ionP r v r

Use the Boltzmann relation:

Simplify the collision terms:

/0 0

ee KTn n e n e

½, / , and ( / )e s ee KT c KT M E

Reduce to one dimension in r

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2 2

2 2 2( ) ( )s

n i n i cs s

cdv v vn P r n P P

dr rc v c

Eliminate n and to get an equation for v(r):

/ , ( ) 1 ( ) / ( )s c iu v c k r P r P r Non-dimensionalize:

22

1(1 )

1n

is

ndu uP ku

dr r cu

This is an ordinary differential equation for all the plasma

profiles.

Rescale r to see structure of the equation

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( / )n i sn P c r

( ) 1 ( ) / ( )c ik r P r P r

22

11

1

du uku

d u

22

1(1 )

1n

is

ndu uP ku

dr r cu

We had:

Rescale r:

Finally:

k contains the plasma information:

Solutions for uniform pressure and KTe

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0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

V /

Cs

a

a

a

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0r / a

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

n/n0

eV/KTe

v/cs

Solutions for three values of k Rescale so that a 1 in each case

This profile is independent of pressure, size, and magnetic field.This profile is independent of pressure, size, and magnetic field.

It depends on It depends on KTKTee, but is always peaked at the center., but is always peaked at the center.

This profile IS modified:

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• When Te is changed or varies with r

• When nn varies with r (neutral depletion, treated later)

• When k varies with r

But the central peaking remains

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0r / a

n /

n0

110100

p (mTorr)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1r / a

n /

n0

234

KTe (eV)

Ionization balance restricts KTe for real r

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1( )n i e

drnv n P T

nr dr

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5r [cm]

V /

Cs

3.00

3.49

4.00

KTe (eV)

22

1(1 )

1n

is

ndu uP ku

dr r cu

Our previous dimensional equation

Solved simultaneously

Improved Te – p0 relation

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0

2

4

6

8

10

1 10 100p0 (mTorr)

KT

e (

eV)

2.5

5

10

Tube radius (cm)

Old, radially averaged data:M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed. (Wiley-Interscience, Hoboken, NJ, 2005).F. F. Chen and J.P. Chang, Principles of Plasma Processing (Kluwer/Plenum, New York, 2002),

The EQM program solves simultaneously:

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22

1( ) 1 (1 / )

1n

i c is

ndu uP r u P P

dr r cu

1( )n i e

drnv n P T

nr dr

2n n iD n n nP

Ion motion

Neutral depletion

Ionization balance

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5r (cm)

n (1

012

cm-3

) &

p

/ p

0

1

5

10

p0 (mTorr)

Last step: iteration with HELIC

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0

2

4

6

8

0.0 0.5 1.0 1.5 2.0 2.5r (cm)

n (1

011

cm

-3)

0

1

2

3

4

5

6

Pr (K

W/m

2)n

Pr

13.56 MHz65G, 400W

0

1

2

3

4

5

6

0.0 0.5 1.0 1.5 2.0 2.5r (cm)

KT

e (e

V)

14.4

14.6

14.8

15.0

15.2

p (mT

orr)

KTe (eV)p (mTorr)

13.56 MHz65G, 400W

Lc

a b

h

Loop antenna

Helical antenna

B0

Another layer off the onion!

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