A new theory of gas dischargesbased on experimentsFrancis F. Chen, University of CaliforniaKAIST, Daejeon, S. Korea, April 2011

ICPs show anomalous skin depthUCLAPlasma density is peaked on axis even when antenna is on periphery.

Helicon discharges are also peaked on axisUCLATypical RF deposition profiles are peaked at edge due to the TG modeTypical center-peaked density profile

First solution to skin depth problem

Consider the nonlinear effect of the Lorentz forceon the motion of an electron in an RF field Eq

This force is in the radial direction, while E is in the q direction

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Problem 2: Diffusion across BClassical diffusion predicts slow electron diffusion across BHence, one would expect the plasma to be negative at the center relative to the edge.

n

B

n

i

e

i

e

B

Density profiles are never hollowIf ionization is near the boundary, the density should peak at the edge. This is rarely observed.

Consider a discharge of moderate length UCLAElectrons are magnetized; ions are not.Neglect axial gradients.Assume Ti

We now have a simple equilibrium problemUCLAIon fluid equation of motionionizationconvection CX collisionsneglect Bneglect TiIon equation of continuitywhereResult

Reduce to 1D and normalizeUCLAResultion motionion continuityAll radial dependences are kept so far.

Electrons: short-circuit effect (1)UCLAAssume ionization is higher on the outside (tube 1)Ions will diffuse inwards, to tube 2Electrons cant follow, but the sheath drop in 2 can increase to trap more electrons to neutralize the ions. Electrons can effectively move across B by the sheath adjustmentTHIS SHEATH ADJUSTMENT TAKES ONLY NANOSECONDS

Electrons: short-circuit effect (2)UCLAThe sheaths adjust to Maxwellian: is high where n is high (tube 1)An electric field Er arises to drive ions inwards.Er is scaled to KTe, so n(r) reaches equilibrium faster than normal.In equilibrium, n(r) is higher in center, so E is reversed to drive ions out .Te(r) cannot be changed by short circuit effect.On a millisecond diffusion time scale

The electrons can then follow BoltzmannUCLA The short-circuit effect allows electrons to thermalize across B in nsecs. Electrons then fall into their most probable distribution: Maxwellian They then follow the Boltzmann relation everywhere.This is our basic assumptionIt has amazing consequences!

3 equations for 3 unknowns: v, n, UCLAion motionion continuityelectron BoltzmannEliminating n and , we have an ODE for v :

Matching to Debye sheath is automaticUCLAThis plasma solution has dv/dr as v cs at the sheath edge.All radial variations can be taken into account in this equation.To better see the structure of this equationIntroduce dimensionless variables

Matching to Debye sheath is automaticUCLAThis equation yields the profiles of v, n, and Vs in equilibrium

Properties of this universal equationUCLA Plasma properties enter only in k (r ) in the nonlinear u 2 term.The equilibrium profiles are the same if k is the same. The neutral density nn does not appear in k. The profiles are unchanged if the quantities in are changed. The B-field does not appear. B can be strong or zero.

Solutions for constant kUCLALet nn and Te be uniform, so k is a constant.Self-similar profiles for 3 values of kRe-scale so a = r /a for one kAll profiles are identical for any pressure or tube radiusAnd n (r ) is always peaked on axis, as long as axial losses are negligible.

No presheath matching is neededNot only are collisions and ionization taken into account everywhere, but Pc(r)s variation is known since vi is known

n

xs

x

ne = ni = n

PLASMA

SHEATH

ni

ne

+

ns

PRESHEATH

v = cs

Dependence on pressure and TeUCLAk is independent of nn, but Pi depends on Te.

Ionization balanceUCLAKnowing the ion velocity vs. r, we can calculate the loss of ions from each radial tube. Ionization must replace them in equilibrium.This gives an inverse relation between nn and Te, which depends on radius a.

Neutral depletionUCLATo keep things simple we have to make some crude approximations.The n0-n0 mfp is short, so the neutrals diffuse.Recycling causes uniform input at r = a at all z.Output at r = a is also uniform because of many bounces before entering pump.

Neutral depletion (2)UCLAAt very high density, nn gets very low, and therefore Te is unreasonably high.This is because ENERGY BALANCE has not been considered.To get Te(r), we have to specify the energy input of a specific discharge.

The EQM codeUCLAWe now have three differential equations:Ion motion, with Maxwellian electrons, in dimensional unitsIonization balance at each rNeutral depletionThe EQM code by Curreli solves all three simultaneously for givenPlasma density on axisInitial neutral densityKTe, either uniform or with given profile

UCLAAn antenna launches what is essentially a whistler wave in a DC magnetic fieldWe now specify the helicon discharge

The HELIC code for helicon dischargesThe 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

Energy balance gives us the data to calculate Te(r)The particle losses Wi and We are simple. The radiation loss Wr, however, has to be calculated from the Vahedi curve, known for argon. This gives the energy needed to create each ionization, including all the radiation that is lost before that happens.Knowing Pin(r), we can calculate KTe from this curve.

Why are helicon dischargessuch efficient ionizers?The helicon wave couples to an edge cyclotron mode, which is rapidly absorbed.

Density profiles for given Pin profilesUCLAThese are power deposition profiles for different cases with uniform n(r), from HELICThese are the corresponging density profiles from EQM, using Te(r) from the Vahedi curve.However, Pin (r) was not calculated from n(r). We must iterate.

Sample of EQM-HELIC iterationUCLAIt takes only 5-6 iterations before convergence.Note that the Tes are now more reasonable.Tes larger than 5 eV reported by others are spurious; their RF compensation of the Langmuir probe was inadequate.

Comparison with experimentUCLAThis 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

Location of downstream probesUCLA

Probe at Port 1, 6.8 cm below tubeUCLAThe density peaks on axisTe shows Trivelpiece-Gould deposition at edge.Vs(Maxw) is the space potential calc. from n(r) if Boltzmann.

Dip at high-B shows failure of modelUCLAWith 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)UCLAThe RF power deposition is not uniform axially, and the equivalent length L of uniform deposition is uncertain within the error curves.

SummaryUCLAThe short circuit effect allows electrons to cross a B-field.Density profiles in equilibrium have almost universal shape.Equilibrium profiles agreeing with experiment have been computed for the first time.This permits simpler design of industrial plasma sources.For helicon discharges, absolute agreement be obtained.