Cylindrical probes are best Plane probes have undefined

Cylindrical probes are best 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 theory is easier. UCLA

In RF plasmas, the probe is more complicated! UCLA

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

The electron characteristic UCLA

The transition region (semilog plot) UCLA

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

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

Thin sheaths: can use Bohm current UCLA

Floating potential (I = 0) Set Isat = Ie. Then Vp = Vf, and UCLA

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

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

Vs at end of exponential part UCLA

Ideal minimum of d. I/d. V UCLA

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

Two ways to connect a probe 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. UCLA

RF Compensation: the problem 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, Vs is ~constant along field lines, so the potential oscillations extend throughout the discharge. UCLA

Effect of Vs osc. on the probe curve UCLA

The I-V curve is distorted by the RF UCLA

Solution: RF compensation circuit* * V. A. Godyak, R. B. Piejak, and B. M. Alexandrovich, Plasma Sources Sci. Technol. 1, 36 (19920. I. D. Sudit and F. F. Chen, RF compensated probes for high-density discharges, Plasma Sources Sci. Technol. 3, 162 (1994) 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) A commercial probe with replaceable tip UCLA

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

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

Equivalent circuit for RF compensation The dynamic sheath capacitance Csh 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) UCLA

Electron distribution functions 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. UCLA

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

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

Cautions about probe EEDFs • 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) UCLA

Summary of ion collection theories (1) 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 Ti, the formula becomes very simple: I 2 varies linearly with Vp (a parabola). This requires thin probes and low densities (large l. D). 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

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

ABR curves for cylinders, Ti = 0 xp = Rp/l. D, hp = Vp/KTe 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 ni twice. UCLA

The Bernstein-Rabinowitz-Laframboise (BRL) theory (2) The ions have a monoenergetic, isotropic distribution at infinity. Here b is Ei/KTe. 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. 113 -200 (1965) UCLA

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

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

Variation of a with xp

Summary: how to measure density with Isat High density, large probe: use Bohm current as if plane probe. Ii does not really saturate, so must extrapolate to floating potential. Intermediate Rp / l. 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 highdensity plasmas, Phys. Plasmas 8, 3029 (2001) UCLA

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

Comparison of various theories (2) UCLA

Comparison of various theories (3) 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. UCLA

Reason: collisions destroy orbiting An orbiting ion loses its angular momentum in a chargeexchange 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

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

Unexpected success of the C-L (Vf) 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

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

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

Direct verification of potential pulling UCLA

Correcting for Vf shift gives better I-V curve UCLA

Hence, we must use a dc reference electrode HERE UCLA

Recent data in Medusa 2 (compact) Probe outside, near wall n = 0. 81 1011 cm-3 Te = 1. 37 e. V UCLA

Recent data in Medusa 2 (compact) Probe under 1 tube, 7” below, 3 k. W, 15 m. Torr n = 3. 12 1011 cm-3 UCLA

Recent data in Medusa 2 (compact) Te (bulk) = 1. 79 e. V Probe under 1 tube, 7” below, 3 k. W, 15 m. Torr Te (beam) = 5. 65 e. V UCLA

Other important probes not covered here • 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! UCLA

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! UCLA