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Slide 2 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. Slide 3 In RF plasmas, the probe is more complicated! UCLA Slide 4 Parts of a probes IV 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 Slide 5 The electron characteristic UCLA Slide 6 The transition region (semilog plot) UCLA Slide 7 The exponential plot gives KT e (if the electrons are Maxwellian) UCLA Slide 8 Ion saturation current UCLA This gives the plasma density the best. However, the theory to use is complicated. Slide 9 Thin sheaths: can use Bohm current UCLA Slide 10 Floating potential (I = 0) UCLA Set I sat = I e. Then V p = V f, and Slide 11 Electron saturation usually does not occur UCLA Ideal for plane Reasons: sheath expansion collisions magnetic fields Slide 12 Extrapolation to find knee and V s UCLA Vs?Vs? Slide 13 V s at end of exponential part UCLA Slide 14 Ideal minimum of d I /dV UCLA Slide 15 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). Slide 16 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. Slide 17 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. Slide 18 Effect of V s osc. on the probe curve UCLA Slide 19 The I-V curve is distorted by the RF UCLA Slide 20 * 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) Solution: RF compensation circuit* UCLA Slide 21 Effect of auxiliary electrode The chokes have enough impedance when the floating potential is as positive as it can get. Slide 22 The Chen B probe UCLA Slide 23 Sample probes (3) UCLA A commercial probe with replaceable tip Slide 24 Sample probes (1) UCLA A carbon probe tip has less secondary emission Slide 25 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. Slide 26 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) Slide 27 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. Slide 28 Examples of non-Maxwellian distributions UCLA EEDF by Godyak A bi-Maxwellian distribution Slide 29 Example of a fast electron beam UCLA The raw dataAfter subtracting the ion current After subtracting both the ions and the Maxwellian electrons Slide 30 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) Slide 31 Summary of ion collection theories (1) UCLA Langmuirs Orbital Motion Limited (OML) theory Integrating over a Maxwellian distribution yields Slide 32 UCLA Langmuirs 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). Slide 33 UCLA Langmuirs Orbital Motion Limited (OML) theory (3) Slide 34 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. Slide 35 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. Slide 36 ABR curves for cylinders, T i = 0 UCLA p = R p / D, p = V p /KT e Slide 37 UCLA Summary of ion collection theories (3) The Bernstein-Rabinowitz-Laframboise (BRL) theory The problem is to solve Poissons equation for V(r) with the ion density depending on their orbits. Those that miss the probe contribute to n i twice. Slide 38 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. 113-200 (1965) Slide 39 UCLA The Bernstein-Rabinowitz-Laframboise (BRL) theory (3) Example of Laframboise curves: ion current vs. voltage for various R p / D Slide 40 UCLA Summary of ion collection theories (4) Improvements to the Bohm-current method Slide 41 Variation of with p Slide 42 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 Slide 43 Parametrization of Laframboise curves UCLA Slide 44 The fitting formulas BRL ABR F.F. Chen, Langmuir probe analysis for high- density plasmas, Phys. Plasmas 8, 3029 (2001) Slide 45 Comparison of various theories (1) UCLA The geometric mean between BRL and ABR gives approximately the right density! Slide 46 Comparison of various theories (2) UCLA Slide 47 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. Slide 48 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 Slide 49 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. Slide 50 UCLA The floating potential method for measuring ion density s cscs VfVf d (Child-Langmuir law) Slide 51 Unexpected success of the C-L (V f ) method Slide 52 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 Slide 53 Ideal OML curve Peculiar I-V curves: not caused by RF Slide 54 Probe electron current can pull V s up if the chamber is not grounded UCLA Slide 55 Direct verification of potential pulling UCLA Slide 56 Correcting for V f shift gives better I-V curve UCLA Slide 57 Hence, we must use a dc reference electrode UCLA HERE Slide 58 Recent data in Medusa 2 (compact) UCLA n = 0.81 10 11 cm -3 T e = 1.37 eV Probe outside, near wall Slide 59 Recent data in Medusa 2 (compact) UCLA Probe under 1 tube, 7 below, 3kW, 15 mTorr n = 3.12 10 11 cm -3 Slide 60 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 Slide 61 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! Slide 62 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! Slide 63