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

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

Post-Langmuir probe theories - 2 UCLA

Post-Langmuir probe theories - 3 UCLA

Post-Langmuir probe theories - 4 UCLA

Probes in fully ionized plasmas

Experimental verification in Q-machine - 1 UCLA

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

Experimental verification in Q-machine - 3 UCLA

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

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

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

The real density is close to the geometric mean! UCLA

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

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

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

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

Introduction: RF distortion of I-V trace - 1 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

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

A large compensation electrode helps UCLA

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

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

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

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

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

Apparatus: anodized walls, floating top plate 1.9 MHz, 60-100W, 3-10 mTorr Ar Ceramic shaft UCLA

Direct verification of potential pulling UCLA

Correcting for Vf shift gives better I-V curve UCLA

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

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

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

Insertion of grounding plate close to probe UCLA

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

But the I-V curves are about the same UCLA

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

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

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

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

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

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

A measured radial density profile UCLA

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

Too fast a scan: sheath not in equilibrium UCLA

Hence we must use a dc reference electrode. HERE UCLA