3. Diagnostics
3.1.2 Langmuir Probe
To find the density of the plasma, a Langmuir probe (LP) is used. In its simplest form a LP is a small biased electrode immersed in the plasma, and which draws a current. If the LP is biased sufficiently negatively, then electrons are repelled from the probe tip, and only positive ions are collected. As the bias is made more negative, the collected current is seen to increase, until a saturation is reached (due to both the absence of electrons, and space charge effects associated with the positive ions), and the current collected no longer increases [112]. When this occurs, the LP is said to be in ion saturation mode (although as will be seen below, a true saturation is in fact not reached) [30]. By sweeping the bias voltage from positive to negative voltages, it is possible to generate a current-voltage (IV) characteristic. This can then be used to find a number of additional plasma properties (such as the electron temperature), but this is not used here and will not be described further.
LP Design
The LP used consists of a small 2 mm diameter nickel disc attached to the end of a hollow ceramic tube. The ceramic tube is then connected to a metal probe shaft, as discussed in the introduction to Section 3.1. The disc is attached to a copper electrical wire (via a physical crimping connection, since normal lead solder will melt at the temperatures the probe is exposed to), which runs through both the ceramic tube and the metal shaft, and is connected to a BNC connector within the back cap. This copper wire is insulated, and a second smaller metal shaft (attached to the larger metal shaft) surrounds this insulation within the ceramic tube, so as to provide a continuous coaxial connection throughout the probe. A cutaway view of the LP probe tip design is shown in Fig. 3.7 (a). The probe tip is biased at−45 V with a number of batteries, and the collected current is found from the voltage drop across a 1 kΩ sense resistor measured with a Digitech QM-1320 multimeter (note that this “averages” out any rf oscillations within the plasma density). Figure 3.7 (b) shows a schematic of the electrical circuit used to measure the plasma current.
Note that with the bias and sense resistor removed from Fig. 3.7 (b), the multimeter only reads a voltage, with no net current flowing. Thus the probe tip floats to a certain voltage relative to ground, which occurs when the ion and electron current to the probe is equal (since no closed current path now exists). This voltage that forms is referred to as the floating potential, and can be used to give approximate indications of the plasma potential and qualitative behaviour of the electron temperature.
Theory of Operation and Data Analysis
The collected current,Isat, is related to the plasma density,n0, from standard LP theory
[30] using
Probe Switch Vbias = -45 V 1 kΩ Multimeter (b) Ceramic tube Metal tube Copper wire Insulation Nickel tip A A B B C C (a)
Fig. 3.7: (a) Sectioned schematic of the LP. The nickel tip is crimped to the copper wire along the region marked C. Region A is the circular disc of the tip, while B represents the stem. (b) Electrical circuit schematic used with the LP. When the switch is closed, the probe tip is biased at
−45 V (provided by a series of 9 V batteries), and the measured current is found from the voltage
across the 1 kΩ sense resistor.
where q is the electron charge, Ap = 2πrp2 is the probe collecting area (the factor of 2 accounts for both sides of the probe tip),rpis the radius of the LP disc, anduB=
p qTe/M is the Bohm velocity withTethe electron temperature, andM the ion mass. As was shown in Fig. 3.7 (a), a nickel probe tip is used for the LP. To prevent plasma from entering the ceramic tube, initially high-temperature ceramic paste, known as Autocrete, was used to seal up the ceramic tube hole, as well as to cover the stem of the nickel disc (thus ensuring the probe collection area is accurately known). However, this Autocrete proved unsatisfactory at high plasma densities (1017 m−3) where the nickel disc was found to
become readily sputtered. This sputtered material would then collect on the Autocrete, and a conductive path could be formed which would change the collecting area (Ap) of the probe tip with time, resulting in inconsistent densities. Figure 3.8 shows an example of a LP illustrating the sputtering and deposition described above. To solve this problem, theAutocrete was removed, and the probe tip was positioned such that the disc stem and electrical wire would “float” within the ceramic tube (so as not to touch the sides of the ceramic tube). In this way any sputtered material would either collect on the exposed metal of the tip itself, or on the inner walls of the ceramic tube, which the probe is now no longer in contact with. Additionally, a switch was added to the measurement circuit (see Fig. 3.7 (b)), so that after a measurement, and while the probe is being repositioned, the bias can be turned off to minimize any sputtering. These changes were found to work very well, and consistent density results were obtained. Unfortunately, the main problem with this method is that the collecting area of the probe tip is now not accurately known. An estimate of the area was made by constructing a second probe using the original design with the Autocrete, and calibrating the new design in a low density plasma with minimal sputtering effects. In this way the new probe was found to have a collecting area about
§3.1 Plasma Diagnostics 65
20−25% larger than the old design.
Autocrete paste
Nickel disc Deposited material
Fig. 3.8: Photograph of a damaged LP tip, indicating the deposited material on the Autocrete
paste. This material comes from the sputtered nickel probe tip, and acts to increase the collecting area of the probe, thus giving inconsistent measurements.
Equation 3.8 is typically valid only for a planar probe tip such as a disc [30, 112], and makes two main assumptions: (1) The plasma electrons have a Maxwellian distribu- tion, and (2) the plasma sheath is small compared to the probe dimensions. The first assumption is in general not true for low pressure plasmas [113], requiring knowledge of the electron distribution function, which is often difficult to determine accurately. Here we continue to use this assumption (which still affords a reasonable estimate of the density to be made as long as the low energy electrons are approximately Maxwellian), but find elec- tron temperatures from the electron distribution function (which can be non-Maxwellian) measured with a compensated LP described in the next section. The second assumption is reasonable for large plasma densities where the sheath is very small, but at the densities used here (1016−1017m−3) this need not be the case. This is especially true for the small probe tip used, since due to edge effects, the collecting area can be much larger than the physical probe area. In fact Ap in Eqn. 3.8 should be the sheath area, not the physical area of the probe [112]. As the probe bias increases, the sheath can undergo expansion, and so the effective collecting area can increase (this therefore means that no true ion saturation current is necessarily reached) [114].
Sheath Expansion
If sheath expansion is not accounted for, then the collecting area of the probe is under- estimated, so that the calculated plasma density is overestimated. Finding the actual collecting area is not simple, especially in the presence of a magnetic field, but can be estimated using a recent model by Sheridan [114, 115]. Sheridan [114] has simulated in detail the flow of plasma to a disc probe operating in ion saturation mode, by using a com-
bination of a fluid description for the plasma electrons, and particle-in-cell ions. In the simulations, by ignoring the probe supporting components (together with magnetic field effects), the effective collecting area of the probe tip is established, accounting for sheath expansion. The results are then fitted to power laws, allowing the effective collecting area to be determined for different probe biases and disc radii. This then provides a simple method to find the actual sheath area, As, which is found from
As Ap
= 1 +aηpb (3.9)
whereηp =−(Vbias−Vp)/Te,Vp andVbias are the plasma potential and probe bias (which is −45 V) respectively, and the coefficientsa andb are given by
a= 2.28ρ−p0.749 (3.10)
b= 0.806ρ−p0.0692 (3.11)
whereρp =rp/λDe, and the electron Debye length is found from
λDe= ǫ0Te qn 1/2 (3.12)
withǫ0the permittivity of free space. The current,Isat, collected by a LP in ion saturation mode is then
I =κqAs(n)nuB (3.13)
where the sheath area is now a function of the plasma density (through Eqns. 3.9 - 3.12), and where κ ≈0.55 following Sheridan’s suggestion [114]. Experimental confirmation of Sheridan’s theory has recently been conducted [116], showing that the sheath expansion is accurately modelled by the simulations. Sheridan estimates that the simulation probably overestimates the sheath area by about 15% at most. This above method does not how- ever account for magnetic field effects, which very little work in the literature addresses comprehensively. For applied magnetic fields to have a small effect on the LP results, the probe tip dimension should be smaller than the gyroradius of electrons and ions present [112]. For the low magnetic fields used in this thesis (B0<0−15 mT), the ion gyroradius radius is greater than 2 cm, much larger than the LP disc, while the electron gyroradius is less than about 1 mm. However, since the LP is operated in ion saturation mode (so that electrons would be repelled anyway), and since the probe tip is orientated perpendic- ular to the magnetic field for almost all measurements, any anisotropy introduced by the
§3.1 Plasma Diagnostics 67
magnetic field should not be too severe.
The error in the measured current from the multimeter is very small, being only about a few percent, so that the main error associated with the plasma density in Eqn. 3.13 (within the context of the assumptions used to arrive at this equation that is) is from the electron temperature (which is found using the compensated LP described in the next section, and typically has an uncertainty of about ±0.5 eV). As an example, for an electron temperature of 5±0.5 eV and a collected current of 0.4 mA, the uncertainty in the calculated plasma density is about±5%, or between−5% and 27% if the possible 15% overestimation of the sheath area from Sheridan’s model is accounted for.