2. Starting point: Langmuir’s OML theory
No integration necessary; very simple formula for ion current.
This requires very small Rp / lD, so that there is no absorption radius.
UCLA

3. Post-Langmuir probe theories - 1
Sheath, but no orbiting
UCLA

4. Post-Langmuir probe theories - 2
UCLA

5. Post-Langmuir probe theories - 3
UCLA

6. Post-Langmuir probe theories - 4
UCLA

7. Probes in fully ionized plasmas

8. Experimental verification in Q-machine - 1
UCLA

9. Experimental verification in Q-machine - 2
Such nice exponentials were never seen again!
UCLA

10. Experimental verification in Q-machine - 3
UCLA

11. 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
Getting good probe data is much more difficult!
UCLA

12. Ion currents in an ICP discharge
They fit the OML theory, which is not applicable!
UCLA

13. Each theory yields a different density
Here
 Rp / lD
UCLA

14. The real density is close to the
geometric mean!
UCLA

15. 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

16. This collisional effect has been verified
Sternovsky, Robertson, and Lampe, Phys. Plasmas 10, 300 (2003).
Sternovsky, Robertson, and Lampe, J. Appl. Phys. 94, 1374 (2003).
Rp/lD = 0.05
Rp/lD = 0.49
Rp/lD = 0.26
The extra ion current due to collisions is calculated and added to the OML current. The result agrees with measurements only for very low density (< 108 cm-3).
The theory is incomplete because the loss of orbiting ions is not accounted for. Also, there is no easy computer program.
UCLA

17. 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

18. 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

19. Introduction: RF distortion of I-V trace - 1
UCLA

20. 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

21. Self-resonance of choke chains
To get high impedance, self-resonance of chokes must be used. Chokes must be individually chosen because of manufacturing variations.
UCLA

22. A large compensation electrode helps
UCLA

23. What is the sheath capacitance as Vs oscillates?
Ideal OML curve
A small RF oscillation will bring the probe from the Child-Langmuir sheath to the Debye sheath to electron saturation
UCLA

24. Sheath capacitance: exact vs. C-L
This is an extension of the work by Godyak:
V.A. Godyak and N. Sternberg, Phys. Rev. A 42, 2299 (1990)
V.A. Godyak and N. Sternberg, Proc. 20th ICPIG, Barga, Italy, 1991, p. 661
UCLA

25. Variation of Csh during an RF cycle
Large probe, which draws enough current to affect Vs.
These curves will give rise to harmonics!
A normal small probe, which goes into electron saturation.
Cylindrical effects will smooth over the dip.
UCLA

26. 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

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

28. Potential pulling by probe
Curves taken with two probes, slowly, point by point
UCLA

29. Apparatus: anodized walls, floating top plate
1.9 MHz, W, 3-10 mTorr Ar
Ceramic shaft
UCLA

30. Direct verification of potential pulling
UCLA

31. Correcting for Vf shift gives better I-V curve
UCLA

32. Slow drift of probe currents: ions
A scan takes 2-3 sec (200 points), and ~3 sec between scans.
The time constant is very long.
UCLA

33. Slow drift of probe currents: electrons
The drift direction depends on the parking voltage between scans.
The drift can continue for >10 sec.
UCLA

34. This is the right order of magnitude.
Reason: the walls are charged through the probe
The only connection to ground is through the probe.
The plasma potential has to follow Vp.
Hence the capacitance of the insulating layer has to be charged.
CV = Q = I*t, t = CV/ I
C = R0Aw/d, Aw = 0.44 m, R ~ 3, d ~ 1 m
C ~ 10 F, V ~ 100 V, Ie ~ 2 mA
 t ~ 0.5 sec
This is the right order of magnitude.
Slower drifts may be due to small leaks in the insulation.
UCLA

35. Insertion of grounding plate close to probe
UCLA

36. Grounding plate reduces change in Vf
High pressure (9.7 mTorr)
Low pressure (2.7 mTorr)
UCLA

37. But the I-V curves are about the same
UCLA

38. Compare with ideal OML curve
The ion part fits well.
The electron part, after correcting for the Vf shift, fits the exponential region better, but still fails at saturation.
The remaining discrepancy must be due to inadequate RF compensation.
UCLA

39. Applying +100V to probe suddenly
SOURCE e + + + + + e + e
WALL + + + e + + e
Vs ~ Vs0 e e e
There is an initial transient, but a normal electron sheath at electron saturation should come to equilibrium in several ion plasma periods (<< 1 msec).
UCLA

40. With a grounding plane, how can a probe affect Vs?
Normally, the probe current Ie is balanced by a slight adjustment of the electron current to the walls, Iew, via a small change in sheath drop. Since Iew = Iiw, Vs should not change detectably if Ie << Iiw.
UCLA

41. Let’s work out the numbers
Bohm current density: Ii = 0.5 neAwcs ( n = 2 x 1010 cm3, KTe = 1.6 eV)
Ion current to grounding plate (25 cm2) » 8.5 mA
Electron saturation current at +100V = 25 mA (measured)
(Same order of magnitude, within variations.)
Thus, at high Vp, ion loss is too small to balance electron loss.
BUT: Vs changes well before Ie reaches 25 mA
The ion flux to ground may be less than Bohm.
UCLA

42. If no grounding plate, how long does it take for the ions to redistribute themselves?
If the probe draws excess electrons at the center, an ambipolar field will develop to drive ions faster to the wall. The density profile n(r) will change from essentially uniform to peaked.
The diffusion equation for a nearly spherical chamber is
where D = Da, the ambipolar diffusion coefficient. The solution is
The time constant for the lowest radial mode j = 1 is then
UCLA

43. Time to change from uniform to peaked profile
Thus, the time required for the ions to adjust to a new equilibrium is only about 1 msec or less.
UCLA

44. A measured radial density profile
UCLA

45. Conclusion: timing is critical
The dwell time must be long enough for the sheath to come into
equilibrium. This is several ion plasma periods (>100 nsec).
The total sweep time must be << 1 msec, or the plasma potential
will change.
With very slow sweeps, Vs will change and must be monitored.
Even a DC, point-by-point measured I-V curve may not be correct.
UCLA

46. Too fast a scan: sheath not in equilibrium
UCLA

47. Hence we must use a dc reference electrode.
HERE
UCLA