PM Helicons, a Better Mouse Trap

UCLA

Part 1: Permanent-magnet helicon sources and arrays

Part 2: Equilibrium theory of helicon and ICP discharges with the short-circuit effect

Commercial helicon sources required large electromagnets

The PMT MØRI helicon etcher

Helicon waves require a DC magnetic field

Helicon waves are excited by an RF antennaand propagate away while creating plasma

Field lines in permanent ring magnets

The strong, uniform field is inside the rings, but the plasma cannot get out. However, the reverse external field extends to infinity.

Place the plasma below the magnet

The B-field is almost straight, and its strength can be changed by moving the magnet.

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

-10 -8 -6 -4 -2 0 2 4 6 8 10

Designing the magnet

It should be small and not heavy.

It is 3 in. ID and 5 in. OD, 1 in. thick.

Material: NdFeB

Internal field: 1.2 T (12 kG)

12.7 cm

7.6 cm

PLASMA

Designing the discharge tube

A long, helical antenna has the best coupling, but plasma is lost to the walls before it is ejected.

A short stubby tube with a short antenna has more efficient ejection.

The final tube design

1. Diameter: 2 inches

2. Height: 2 inches

3. Aluminum top

4. Material: quartz

5. “Skirt” to prevent eddy currents canceling the antenna current

5.1 cm

10 cm

5 cmANTENNA

GAS INLET (optional)

The antenna is a simple 3-turn loop, not a helix. The loop antenna must be as close to the exit aperture as possible.

Reflections from the endplates are included.

The power absorption distribution is calculated, as well as the total plasma resistance including the

strong TG mode absorption at the boundary.

Lc

a b

h

Loop antenna

Helical antenna

B0

Designing the helicon wave: the HELIC code

Trivelpiece-Gould mode

Helicon mode

written by Don Arnush

For small tubes, we must use the low-field peak effect

The plasma loading must overcome the circuit losses Rc:

Rp >> Rc

This is helped by adjusting the backplate position so that the reflected

backward wave interferes constructively with the forward wave.

This effect is important here because the external field of permanent

magnets is relatively weak.

This condition for good coupling sets the height of the tube

Designing the discharge

B is uniform in r but doubles in z within tube

0

50

100

150

200

250

300

0 2 4 6 8 10 12z (in.)

Bz (

G)

0.00.50.9

radius (in.)

D

B ~ 60G at D = 6”, antenna to magnet midplane

16 cm

12.7 cm

HELICON ARRAYS

Uniformity with discrete sources was proved long ago

Density uniform to +/- 3%

ROTATING PROBE ARRAY

PERMANENT MAGNETS

DC MAGNET COIL

Two kinds of arrays were tried

Staggered array

Covers large area uniformly for substrates moving in the y-direction

165 cm

53.3 cm

17.8

17.8

17.8

35.6 cm

x

y

165 cm

53.3 cm

17.8

17.8

17.8

x

yTop view 17.8 Compact array

Gives higher density, but uniformity suffers from

end effects.

How did we choose the spacing between tubes?

10 mTorr Ar, 500W @ 13.56 MHz, Z2 @ 18 cm below tube

0

1

2

3

4

5

-20 -10 0 10 20r (cm)

n (1

011 c

m-3

)

34 cm

36 cm

D

Z1

Z2

Pyrex tube, 2-in. diam

Ceramic magnet stack

Radial profile of a single tube

How did we choose the spacing between tubes?

The tube spacing is chosen so that the ripple is less than +/- 2%

Z-2 Z-1 Z2

y

Z

r

y

yLd

Z1

d

0

2

4

6

8

10

12

15 20 25 30L (cm)

Rip

ple

(+/-

%)

0

1

2

3

4

5

6

7

8

-30 -20 -10 0 10 20 30x (cm)

n (1

011 c

m-3

)

y = 7.5

Average over all y

The matching circuit

The problem is that all the Z2 cables must be the same length, and they cannot all be short if the tubes are far apart. The cable lengths and

antenna inductance L are constrained by the matching circuit.

R, L

R, L

R, L

R, L

PS

N loads

Z2 - short cables

Distributor

Z1Z2

Z1 - long cable

C1C2

Matching ckt. 50W

Design of the Medusa 2 machine

65"

21"

7"

7"

7"

7"29"

3.5"

12"7"

3/4” aluminum

Endplates: gas feed and probe port at each end.

Tubes set in deeply (½ inch)

1/2" aluminum

Side view

165 cm

30 cm

15 cm

Probe ports

Aluminum sheetHeight can be adjusted electrically if desired

The source requires only 6” of vertical space above the process chamber

Z1Z2

Operation with cables and wooden magnet tray

Detail of a discharge tube for 13.56 MHz

3-turn antenna of 1/8” copper tubing

Rectangular transmission line

50-W line with ¼” diam Cu pipe for cooled center conductor

Operation with rectangular transmission line

Radial profile between tubes at Z2

0

0.5

1

1.5

2

2.5

3

3.5

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

n (1

011 c

m-3

) nKTe

Density profiles along the chamber

Staggered configuration, 2kW

Bottom probe array

0

1

2

3

4

5

-8 -6 -4 -2 0 2 4 6 8 10 12 14 16x (in.)

n (1

011 c

m-3

)

-3.503.5

Staggered, 2kW, D=7", 20mTorr

y (in.)

UCLA

Density profiles along the chamber

Compact configuration, 3kW

Bottom probe array

0

2

4

6

8

10

-8 -6 -4 -2 0 2 4 6 8 10 12 14 16x (in.)

n (1

011 c

m-3

)

3.5-03.5

Compact, 3kW, D=7", 20mTorr

y (in)

Data by Humberto Torreblanca, Ph.D. thesis, UCLA, 2008.

Part 2

Equilibrium theory of helicon and ICP

discharges with the short-circuit effect

Motivation: Why are density profiles never hollow?

In many discharges, ionization is at the edge. Yet, the plasma density is always peaked at the center.

0

z (c

m)

PlasmaTherm ICP Source Module on top of Magnetic Bucket

Case of an ICP: B = 0

0

2

4

6

8

10

12

-5 0 5 10 15 20r (cm)

n (1

010

cm

-3)

800240200

Prf(W)3 mTorr, 1.9 MHz

Helicon case: Trivelpiece-Gould ionization at edge

Typical radial energy deposition profiles as computed by HELIC

0

200

400

600

800

1000

1200

0 0.5 1 1.5 2 2.5r (cm)

Pr (

W/m

2 )

Case 1Case 2Case 3

First data (1991) in long, 5-cm diam tube. Since then, all n(r)’s have peaked on axis.

Classical diffusion prevents electrons from crossing B

If ionization is near the boundary, the density profile would be hollow. This is never observed.

B

n

r

The Simon short-circuit effect cures this

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

HIGH DENSITY

LOWER DENSITY

SHEATHB

+

+e

e

APPARENT ELECTRON FLOW ION DIFFUSION

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 een dn dr

This radial electric field pushes ions from high density to low density, filling in hollow profiles so that, in equilibrium, the density is always

peaked on axis.

An idealized model of a high-density discharge

1. B = 0 or B 0; it doesn’t matter.

2. The discharge is cylindrical, with endplates.

3. All quantities are uniform in z and θ (a 1-D problem).

4. The ions are unmagnetized, with large Larmor radii.

5. Ion temperature is negligible: Ti / Te = 0

L

aB

+

rLi >> a

rLe << a-

The ion equations

Motion: (Mn en v v E v B ) iKT ion Mn v

½, / , and ( / )e s ee KT c KT M E

2vv vrr s io rd dcdr dr

( ) ( ) ( )n in Q r nn P r vContinuity: ( ) ( )i ionP r v r

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

d d n n P rdr dr r

1-D:

Pi is the ionization probability

( )io n cx n cn v n P r

Pc is the collision probabilityUnknowns: vr(r) (r) n(r)

Combine 2 ion equations with electron Boltzmann relationTo eliminate unknowns (r) and n(r)

2 2

2 2 2 ( ) ( ) 0sn c n i

s s

cdv v v n P r n P rdr rc v c

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 / /c i cx ionk P P v v

We next define dimensionless variables

to obtain…

( )rv v

The universal equation

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

dimensions of 1/r, so we can define

( / )n i sn P c r

22

1 1 01

n i

s

n Pdu ukudr c ru

This yields 22

1 11

du ukud 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.

Solutions for different values of k = Pc / Pi

Identifying a with the discharge radius a gives the same curves in all cases.

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

Since v(r) is known, the collision rate can be calculated with the exact v(r) at each radius. No presheath assumption is necessary.

Further developments in the theory

1. Arbitrary ionization profiles. EQM code

2. Ionization balance: Te(r) depends on p(r)

3. Neutral depletion with velocity-dependent Pc

4. Iteration with HELIC to get ionization profiles for helicon discharges which are consistent with the Te and p profiles from EQM.

5. Energy balance using the Vahedi curve to get absolute densities for given antenna power

This work was done with Davide Curelli, University of Padua

There is absolute agreement with experiment

Single-tube apparatus with probe inside the source

PERMANENT MAGNET

GAS FEED

HEIGHT ADJUSTMENT

LANGMUIR PROBE

0

2

4

6

8

10

12

0 100 200 300 400RF power (watts)

Den

sity

(1011

cm

-3)

MeasuredCalc. L=20Calc. L=25Calc. L=30

Absolute-value agreement with theory with no adjustable paramters

15 mTorr Ar, 400W @ 13.56 MHz, 65 G

The understanding of helicons never ends, but this is…

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