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 antenna and 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.

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)

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 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. Designing the helicon wave: the HELIC code 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 B ~ 60G at D = 6”, antenna to magnet midplane

HELICON ARRAYS

Uniformity with discrete sources was proved long ago Density uniform to +/- 3%

Two kinds of arrays were tried Staggered array Covers large area uniformly for substrates moving in the y-direction Compact array Gives higher density, but uniformity suffers from end effects.

How did we choose the spacing between tubes? 10 mTorr Ar, MHz, 18 cm below tube 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%

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.

Design of the Medusa 2 machine

Side view Probe ports Aluminum sheet Height can be adjusted electrically if desired The source requires only 6” of vertical space above the process chamber Z1 Z2

Operation with cables and wooden magnet tray

Detail of a discharge tube for 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

Density profiles along the chamber Staggered configuration, 2kW Bottom probe array

UCLA Density profiles along the chamber Compact configuration, 3kW Bottom probe array 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. Case of an ICP: B = 0

Helicon case: Trivelpiece-Gould ionization at edge Typical radial energy deposition profiles as computed by HELIC 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.

The Simon short-circuit effect cures this A small adjustment of the sheath drop allows electrons to “cross the field”.

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. 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: T i / T e = 0

The ion equations Motion: Continuity: 1-D: P i is the ionization probability P c is the collision probability Unknowns: v r (r)  (r) n(r)

Combine 2 ion equations with electron Boltzmann relation To eliminate unknowns  (r) and n(r) This yields an ODE for the ion radial fluid velocity: Note that dv/dr   at v = c s (the Bohm condition, giving an automatic match to the sheath We next define dimensionless variables to obtain…

The universal equation Note that the coefficient of (1 + ku 2 ) has the dimensions of 1/r, so we can define This yields Except for the nonlinear term ku 2, this is a universal equation giving the n(r), T e (r), and  (r) profiles for any discharge and satisfies the Bohm condition at the sheath edge automatically.

Solutions for different values of k = P c / P i Identifying  a with the discharge radius a gives the same curves in all cases. 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: T e (r) depends on p(r) 3.Neutral depletion with velocity-dependent P c 4.Iteration with HELIC to get ionization profiles for helicon discharges which are consistent with the T e 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 Absolute-value agreement with theory with no adjustable paramters 15 mTorr Ar, MHz, 65 G

The understanding of helicons never ends, but this is…

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