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Cylindrical probes are best

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.

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Cylindrical probes are best

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  1. 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 the theory is easier. UCLA

  2. In RF plasmas, the probe is more complicated! UCLA

  3. 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

  4. The electron characteristic UCLA

  5. The transition region (semilog plot) UCLA

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

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

  8. Thin sheaths: can use Bohm current UCLA

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

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

  11. Extrapolation to find “knee” and Vs Vs? UCLA

  12. Vs at end of exponential part UCLA

  13. Ideal minimum of dI/dV UCLA

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

  15. 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

  16. 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, Vsis ~constant along field lines, so the potential oscillations extend throughout the discharge. UCLA

  17. Effect of Vs osc. on the probe curve UCLA

  18. The I-V curve is distorted by the RF UCLA

  19. 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

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

  21. The Chen B probe UCLA

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

  23. Sample probes (1) A carbon probe tip hasless secondary emission UCLA

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

  25. 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

  26. 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

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

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

  29. 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

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

  31. Langmuir’s Orbital Motion Limited (OML) theory (2) For s >> a and small Ti, the formula becomes very simple: I2 varies linearly with Vp (a parabola). This requires thin probes and low densities (large lD). UCLA

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

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

  34. 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

  35. ABR curves for cylinders, Ti = 0 xp = Rp/lD, hp = Vp/KTe UCLA

  36. 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

  37. 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

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

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

  40. Variation of a with xp

  41. 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 / lD: 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

  42. Parametrization of Laframboise curves UCLA

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

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

  45. Comparison of various theories (2) UCLA

  46. 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

  47. 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

  48. 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

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

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