Two problems with gas discharges 1.Anomalous skin depth in ICPs 2.Electron diffusion across magnetic fields Problem 1: Density does not peak near the.

Two problems with gas discharges

1. Anomalous skin depth in ICPs

2. Electron diffusion across magnetic fields

0

2

4

6

8

10

12

-5 0 5 10 15 20r (cm)

n (

101

0 c

m-3

)

800

240

200

Prf(W)3 mTorr, 1.9 MHz

Problem 1: Density does not peak near the antenna (B = 0)

Problem 2: Diffusion across B

B

nn

i

ei e

B

Classical diffusion predicts slow electron diffusion across B

2 21 ( / )ci ce

c

DD

Hence, one would expect the plasma to be negative at the center relative to the edge.

Density profiles are almost never hollow

If ionization is near the boundary, the density should peak at the edge. This is never observed.

B

n

r

Consider a discharge of moderate length

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1. Electrons are magnetized; ions are not.

2. Neglect axial gradients.

3. Assume Ti << Te

B

a

ION

ELECTRON

Sheaths when there is no diffusion

HIGH DENSITY

LOWER DENSITY

SHEATH

B

+

e

e

+

Sheath potential drop is same as floating potential on a probe.

This is independent of density, so sheath drops are the same.

The Simon short-circuit effect

Step 1: nanosecond time scale

Electrons are Maxwellian along each field line, but not across lines.

A small adjustment of the sheath drop allows electrons to “cross the field”.

This results in a Maxwellian even ACROSS field lines.

HIGH DENSITY

LOWER DENSITY

SHEATH

B

+

+e

e

APPARENT ELECTRON FLOW ION DIFFUSION 1

2

(a)

Sheath drops change, E-field develops

Ions are driven inward fast by E-field

HIGH DENSITY

LOWER DENSITY

SHEATH

B

+

+

e

e

ION DIFFUSION+

-

E

(b)

1

2

The Simon short-circuit effect

Step 2: 10s of msec time scale

The Simon short-circuit effect

Step 3: Steady-state equilibrium

Density must peak in center in order for potentialto be high there to drive ions out radially.

Ions cannot move fast axially because Ez is smalldue to good conductivity along B.

BSHEATH

+

LOWER DENSITY 2

e

+

HIGHDENSITY 1

e

+ e LOWEST DENSITY 3

ION DIFFUSION

+

-

E

Hence, the Boltzmann relation holds even across B

As long as the electrons have a mechanism that allows them to reach their most probable

distribution, they will be Maxwellian everywhere.

This is our basic assumption.

/0 0

( / )( / )e

ee KT

rE KT

n n e n e

en dn dr

We now have a simple equilibrium problem

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

Ion fluid equation of motion

ionization

convection

CX collisions

neglect B

neglect Ti

Ion equation of continuity ( ) ( ) /

( ) ( )c i cx io n

i e ion

P r v r n

P r v r

( ) ( )n in nn P r v wher

e

Result

( ) 0n i cM e Mn P P v v E v

The r-components of three equations

Ion equation of motion:

Ion equation of continuity:

v v(ln )v ( )r rr n i

d d nn P r

dr dr r

3 equations for 3 unknowns: vr(r) (r) n(r)

/0 0

ee KTn n e n e

( / )( / )erE KT en dn drElectron

Boltzmann relation:

(which comes from)

2 ( )s n c idv dv c n P P vdr dr

Eliminate (r) and n(r) to get an equation for the ion vr

This yields an ODE for the ion radial fluid velocity:

Note that dv/dr at v = cs (the Bohm condition, giving an automatic match to the sheath

/ su v c

We next define dimensionless variables

to obtain…

( )rv v2 2

2 2 2( ) ( )s

n i n i cs s

cdv v vn P r n P P

dr rc v c

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

We obtain a simple equation

Note that the coefficient of (1 + ku2) has the

dimensions of 1/r, so we can define

( / )n i sn P c r

This yields 22

11

1

du uku

d u

Except for the nonlinear term ku2, this is a universal equation giving

the n(r), Te(r), and (r) profiles for any discharge and satisfies the

Bohm condition at the sheath edge automatically.

22

1(1 )

1n

is

ndu uP ku

dr r cu

Reminder: Bohm sheath criterion

n

xs x

ne = ni = n

PLASMA

SHEATH

ni

ne

+

ns

PRESHEATH

v = cs

Solutions for different values of k = Pc / Pi

We renormalize the curves, setting a in each case to r/a, where a is the discharge radius. No presheath assumption is needed.

We find that the density profile is the same for all plasmas with the same k.

Since k does not depend on pressure or discharge radius, the profile is “universal”.

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

A universal profile for constant k

These are independent of magnetic field!

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)

k does not vary with p k varies with Te

These samples are for uniform p and Te

Ionization balance and neutral depletion

1( )n i e

drnv n P T

nr dr

2n n iD n n nP

22

10

1n

c is

ndu uu P P

dr r cu

Ionization balance

Neutral depletion

Ion motion

The EQM code (Curreli) solves these three equations simultaneously, with all quantities varying with radius.

Three differential equations

Energy balance: helicon discharges

To implement energy balance requires specifying the type of discharge. The HELIC program for helicons and ICPs can calculate the power deposition Pin(r) for given n(r), Te(r) and nn(r) for various

discharge lengths, antenna types, and gases. However, B(z) and n(z) must be uniform. The power lost is given by

out i e rP W W W

Energy balance: the Vahedi curve

This curve for radiative losses vs. Te gives us absolute values.

10

100

1000

1 10KTe (eV)

Ec

(eV

)1.6123exp(3.68/ )c eVE T

2 5

Energy balance gives us the data to calculate Te(r)

Helicon profiles before iteration

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5r (cm)

Pr (

/m

2)

Case 1

Case 2

Case 3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.5 1 1.5 2 2.5r (cm)

n/n

0

Case1

Case 2

Case 3

Trivelpiece-Gould deposition at edge

Density profiles computed by EQM

These curves were for uniform plasmas

We have to use these curves to get better deposition profiles.

Sample of EQM-HELIC iteration

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0

4

8

12

16

20

0.0 0.5 1.0 1.5 2.0 2.5r (cm)

n (1

011

cm

-3)

0

2

4

6

8

10

12

Pr (k

/m2)

n

Pr

27.12 MHz120G, 1000W

0

1

2

3

4

5

6

0.0 0.5 1.0 1.5 2.0 2.5r (cm)

KT

e (e

V)

14.0

14.2

14.4

14.6

14.8

15.0

15.2

p (mT

orr)

Te (eV)p (mTorr)

27.12 MHz120G, 1000W

It takes only 5-6 iterations before convergence.

Note that the Te’s are now more reasonable.

Te’s larger than 5 eV reported by others are spurious; their RF compensation of the Langmuir probe was inadequate.

Comparison with experiment

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PERMANENT MAGNET

GAS FEED

HEIGHT ADJUSTMENT

LANGMUIR PROBE This is a permanent-magnet helicon source with the plasma tube in the external reverse field of a ring magnet.

It is not possible to measure radial profiles inside the discharge. We can then dispense with the probe ex-tension and measure downstream.

2 inches

Probe at Port 1, 6.8 cm below tube

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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

18V

s (V)

n11KTeVsVs(Maxw)

65 Gauss

1. The density peaks on axis

2. Te shows Trivelpiece-Gould deposition at edge.

3. Vs(Maxw) is the space potential calc. from n(r) if Boltzmann.

Dip at high-B shows failure of model

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0

1

2

3

4

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

n (1

01

1/c

m3),

KT

e (

eV)

KTe

n11

280 Gauss

With two magnets, the B-field varies from 350 to 200G inside the source.

The T-G mode is very strong at the edge, and plasma is lost axially on axis. The tube is not long enough for axial losses to

be neglected.

Example of absolute agreement of n(0)

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0

2

4

6

8

10

12

0 100 200 300 400

RF power (watts)

Den

sity

(10

11 c

m-3

)

Measured

Calc. L=20

Calc. L=25

Calc. L=30

The RF power deposition is not uniform axially, and the equivalent length L of uniform deposition is uncertain within the

error curves.