Cylindrical probes are best UCLA Plane probes have undefined collection area. If the sheath area stayed the same, the Bohm current would give the ion density. A guard ring would help. A cylindrical probe needs only a centering spacer. A spherical probe is hard to make, though the theory is easier.

In RF plasmas, the probe is more complicated! UCLA

Parts of a probe’s I–V curve UCLA V f = floating potential V s = space (plasma) potential I sat = ion saturation current I esat = electron saturation current I here is actually –I (the electron current collected by the probe

The electron characteristic UCLA

The transition region (semilog plot) UCLA

The exponential plot gives KT e (if the electrons are Maxwellian) UCLA

Ion saturation current UCLA This gives the plasma density the best. However, the theory to use is complicated.

Thin sheaths: can use Bohm current UCLA

Floating potential (I = 0) UCLA Set I sat = I e. Then V p = V f, and

Electron saturation usually does not occur UCLA Ideal for plane Reasons: sheath expansion collisions magnetic fields

Extrapolation to find “knee” and V s UCLA Vs?Vs?

V s at end of exponential part UCLA

Ideal minimum of d I /dV UCLA

Finally, we can get V s from V f UCLA V f and KT e are more easily measured. However, for cylindrical probes, the normalized V f is reduced when the sheath is thick. F.F. Chen and D. Arnush, Phys. Plasmas 8, 5051 (2001).

Two ways to connect a probe UCLA Resistor on the ground side does not see high frequencies because of the stray capacitance of the power supply. Resistor on the hot side requires a voltage detector that has low capacitance to ground. A small milliammeter can be used, or a optical coupling to a circuit at ground potential.

RF Compensation: the problem UCLA As electrons oscillate in the RF field, they hit the walls and cause the space potential to oscillate at the RF frequency. In a magnetic field, V s is ~constant along field lines, so the potential oscillations extend throughout the discharge.

Effect of V s osc. on the probe curve UCLA

The I-V curve is distorted by the RF UCLA

* V.A. Godyak, R.B. Piejak, and B.M. Alexandrovich, Plasma Sources Sci. Technol. 1, 36 ( I.D. Sudit and F.F. Chen, RF compensated probes for high-density discharges, Plasma Sources Sci. Technol. 3, 162 (1994) Solution: RF compensation circuit* UCLA

Effect of auxiliary electrode The chokes have enough impedance when the floating potential is as positive as it can get.

The Chen B probe UCLA

Sample probes (3) UCLA A commercial probe with replaceable tip

Sample probes (1) UCLA A carbon probe tip has less secondary emission

Example of choke impedance curve UCLA The self-resonant impedance should be above ~200K  at the RF frequency, depending on density. Chokes have to be individually selected.

Equivalent circuit for RF compensation UCLA The dynamic sheath capacitance C sh has been calculated in F.F. Chen, Time-varying impedance of the sheath on a probe in an RF plasma, Plasma Sources Sci. Technol. 15, 773 (2006)

Electron distribution functions UCLA If the velocity distribution is isotropic, it can be found by double differentiation of the I-V curve of any convex probe. (A one-dimensional distribution to a flat probe requires only one differentiation) This applies only to the transition region (before any saturation effects) and only if the ion current is carefully subtracted.

Examples of non-Maxwellian distributions UCLA EEDF by Godyak A bi-Maxwellian distribution

Example of a fast electron beam UCLA The raw dataAfter subtracting the ion current After subtracting both the ions and the Maxwellian electrons

Cautions about probe EEDFs UCLA Commercial probes produce smooth EEDF curves by double differentiations after extensive filtering of the data. In RF plasmas, the transition region is greatly altered by oscillations in the space potential, giving it the wrong shape. If RF compensation is used, the I-V curve is shifted by changes in the floating potential. This cancels the detection of non-Maxwellian electrons! F.F. Chen, DC Probe Detection of Phased EEDFs in RF Discharges, Plasma Phys. Control. Fusion 39, 1533 (1997)

Summary of ion collection theories (1) UCLA Langmuir’s Orbital Motion Limited (OML) theory Integrating over a Maxwellian distribution yields

UCLA Langmuir’s Orbital Motion Limited (OML) theory (2) For s >> a and small T i, the formula becomes very simple: I 2 varies linearly with V p (a parabola). This requires thin probes and low densities (large lD).

UCLA Langmuir’s Orbital Motion Limited (OML) theory (3)

UCLA Summary of ion collection theories (2) The Allen-Boyd-Reynolds (ABR) theory SHEATH The sheath is too thin for OML but too thick for v B method. Must solve for V(r). The easy way is to ignore angular momentum.

UCLA Allen, Boyd, Reynolds theory: no orbital motion Absorption radius This equation for cylinders was given by Chen (JNEC 7, 47 (65) with numerical solutions.

ABR curves for cylinders, T i = 0 UCLA  p = R p / D,  p = V p /KT e

UCLA Summary of ion collection theories (3) The Bernstein-Rabinowitz-Laframboise (BRL) theory The problem is to solve Poisson’s equation for V(r) with the ion density depending on their orbits. Those that miss the probe contribute to n i twice.

UCLA The Bernstein-Rabinowitz-Laframboise (BRL) theory (2) The ions have a monoenergetic, isotropic distribution at infinity. Here  is E i /KT e. The integration over a Maxwellian ion distribution is extremely difficult but has been done by Laframboise. F.F. Chen, Electric Probes, in "Plasma Diagnostic Techniques", ed. by R.H. Huddlestone and S.L. Leonard (Academic Press, New York), Chap. 4, pp (1965)

UCLA The Bernstein-Rabinowitz-Laframboise (BRL) theory (3) Example of Laframboise curves: ion current vs. voltage for various R p / D

UCLA Summary of ion collection theories (4) Improvements to the Bohm-current method

Variation of  with  p

Summary: how to measure density with I sat High density, large probe: use Bohm current as if plane probe. Ii does not really saturate, so must extrapolate to floating potential. Intermediate Rp / D : Use BRL and ABR theories and take the geometric mean. Small probe, low density: Use OML theory and correct for collisions. Upshot: Design very thin probes so that OML applies. There will still be corrections needed for collisions. UCLA

Parametrization of Laframboise curves UCLA

The fitting formulas BRL ABR F.F. Chen, Langmuir probe analysis for high- density plasmas, Phys. Plasmas 8, 3029 (2001)

Comparison of various theories (1) UCLA The geometric mean between BRL and ABR gives approximately the right density!

Comparison of various theories (2) UCLA

Comparison of various theories (3) UCLA Density increases from (a) to (d) ABR gives more current and lower computed density because orbiting is neglected. BRL gives too small a current and too high a density because of collisions.

Reason: collisions destroy orbiting An orbiting ion loses its angular momentum in a charge- exchange collision and is accelerated directly to probe. Thus, the collected current is larger than predicted, and the apparent density is higher than it actually is. UCLA

Including collisions makes the I - V curve linear and gives the right density Z. Sternovsky, S. Robertson, and M. Lampe, Phys. Plasmas 10, 300 (2003).\ However, this has to be computed case by case.

UCLA The floating potential method for measuring ion density s cscs VfVf d (Child-Langmuir law)

Unexpected success of the C-L (V f ) method

Problems in partially ionized, RF plasmas Ion currents are not as predicted Electron currents are distorted by RF The dc plasma potential is not fixed UCLA

Ideal OML curve Peculiar I-V curves: not caused by RF

Probe electron current can pull V s up if the chamber is not grounded UCLA

Direct verification of potential pulling UCLA

Correcting for V f shift gives better I-V curve UCLA

Hence, we must use a dc reference electrode UCLA HERE

Recent data in Medusa 2 (compact) UCLA n = 0.81  cm -3 T e = 1.37 eV Probe outside, near wall

Recent data in Medusa 2 (compact) UCLA Probe under 1 tube, 7” below, 3kW, 15 mTorr n = 3.12  cm -3

Recent data in Medusa 2 (compact) UCLA T e (bulk) = 1.79 eV T e (beam) = 5.65 eV Probe under 1 tube, 7” below, 3kW, 15 mTorr

Other important probes not covered here UCLA Double probes (for ungrounded plasmas) Hot probes (to get space potential) Insulated probes (to overcome probe coating) Surface plasma wave probes (to overcome coating) Conclusion There are many difficulties in using this simple diagnostic in partially ionized RF plasmas, especially magnetized ones, but the problems are understood and can be overcome. One has to be careful in analyzing probe curves!

Conclusion UCLA There are many difficulties in using this simple diagnostic in partially ionized RF plasmas, especially magnetized ones, but the problems are understood and can be overcome. One has to be careful in analyzing probe curves!