## Sections

Determining Electron Temperature and Plasma Potential

An Additional Concern for Temperature Analysis

This is a simple example of Langmuir probe analysis and the issues related to it. It is intended to serve as a helpful reminder of the technical details in analyzing the data from a swept Langmuir probe and is not a complete theoretical effort. If you have already familiarized yourself with Langmuir probe theory, then you may find this treatment helpful. In this example I begin with the data acquired by measuring the current drawn by a Langmuir probe as the bias applied to that probe is varied. This data is analyzed in order to determine the plasma density, temperature, and potential. While the concept of Langmuir probe usage, digitization of the received data, and even the engineering of the diagnostic are all worthy of discussion, the following has a focus on analysis in order to limit the size of this entry. The data presented in this example was obtained in the undergraduate plasma laboratory (PHYS 180E) at UCLA while I was testing the equipment and helping in the design of the lab exercise as a teaching assistant for the course. Figure 1 represents the circuit used to acquire the signals that will be processed.

An example of what the raw data may look like is provided in figure 2. This data is obtained from an oscilloscope that averages 16 separate acquisitions (while the plasma is continuous, CW, the probe bias sweep is made at a rate of 4 Hz, thereby allowing for multiple acquisitions to be averaged over for one final result). The x-axis units represent the data point number (i.e. if this data was in a spreadsheet, then data point 1000 is the one-thousandth data point you have). The trace labeled V_{bias} represents the applied voltage to the probe. For a properly set up oscilloscope, this signal will be output in the correct units and calibration. The other trace, V_{R}, represents the voltage measured across a resistor in series with the probe. For a given resistor, R, the current through it, I_{probe} (named because it is also passing through the probe) is found using V = I_{probe}R.

Notice that the plot in figure 2 shows a transient effect in the V_{R} trace for values near data index of zero. Be sure to extract only the “proper” portion of these traces when performing your analysis. Zooming in on the data will reveal that we should only consider points 20 through 2500. The points prior to number 20 are an artifact of the bias voltage turn-on. Another striking characteristic of the plot is the stepping feature of V_{R} and to a lesser extent also of V_{bias}. This is not a plasma physics result, rather, the V_{bias} power supply achieves its sweep by stepping the potential up over time. This sweeper is typically operated at a frequency in the kilohertz range and the steps are very difficult to observe. In the 180E lab the plasma is steady state and the sweep frequency has been lowered as much as possible. The sweep rate used is 4 Hz, which results in the individual steps being more noticeable.

It is necessary to convert the V_{R} signal into units of current. Since the electron temperature, T_{e}, trace involves derivatives of the current we must process these raw data in multiple ways. To convert V_{R} into I_{probe} we can use I_{probe} = V_{R}/R. For a resistor of R = 677 Ω this becomes I_{probe} = 1.4771 × 10^{-3} V_{R}.

## Processing the Raw Data

Figure 3 presents a first look at the actual IV trace. This is probe current plotted as a function of probe bias. This plot can tell us a lot about the plasma. The floating potential, V_{f}, occurs where I_{probe} = 0, which appears to give V_{f} ≈ -40 V. The ion saturation current, I_{sat}, is seen at biases well below V_{f}. The roll-off at large positive values of V_{bias} (technically, in this case the largest values of V_{bias} approach zero and may not actually go positive) corresponds to the electron saturation current, e_{sat}. The location of this roll-off, or knee, is the plasma potential, Φ_{p}. As with the floating potential, some estimate of this value may be made from the plot, but a more accurate method will be used to determine the final value. Figure 3 displays Φ_{p} ≈ -15 V.

For swept Langmuir traces such as those presented here, the most important relation between the measurement and plasma parameters is given by,

where q is the electron charge, k_{B} is Boltzmann’s constant, and the constant term will not be important. Notice that this relationship is in the form of a line given by a function f such that f(V) = mx + b, where m is the slope of the line and b is the y-intercept. If our x is actually x = V_{bias} – V_{f}, then the slope of this line is related to the electron temperature. Our method is to plot the term ln|I_{probe} – I_{sat}| and then fit a line to it. The slope of this best fit is inversely proportional to the electron temperature.

It is necessary to subtract the value of the ion saturation current from I_{probe} in order to continue with the analysis. A closer look at figure 3 provides I_{sat} = 1.73 × 10^{−4} A. It is important to get an accurate value of I_{sat} so be sure to read its value careful (i.e. not from a wide range plot like that shown in figure 3). Since the I_{sat} value is negative, subtracting it from I_{probe} will result in almost all of the values of I_{probe} − I_{sat} being positive. This is the intention because we will be working with the natural logarithm of the current in the next few steps and the natural log is not defined for negative numbers. If you have a few negative values left in your trace after subtracting the I_{sat} value that is acceptable. Those values will not play a role in the temperature calculation that follows. Figure 4 shows the resultant electron current with respect to the total current.

## Determining Electron Temperature and Plasma Potential

Figure 5 represents the logarithmic plotting of the electron current. The electron current is I_{probe} − I_{sat} because we have removed the ion contribution to the total current by subtracting I_{sat}. The plot decays very rapidly for values of V_{bias} < V_{f}. This is because ln(0) = -∞ and by subtracting the value of I_{sat} we have forced the current trace to be near zero for values of V_{bias} < V_{f}. The part of this trace that we are interested in occurs above the floating potential and this is the region presented in the next plot.

Figure 6 is a zoomed in version of figure 5 that also includes linear fits to the electron saturation current and inverse temperature. The temperature fit is performed over the range −22 < V_{bias} < -17 V, which can be seen as the exponentially rising region in figure 4. It is important to choose a bias range over which to perform this fit that represents the temperature dependent increase in probe current. By overplotting your data with the linear fit it is possible to quickly demonstrate whether the correct temperature region has been identified. According to this temperature fit, the inverse temperature is approximately 0.27, which leads to T_{e} = 3.70 eV.

A linear fit to the electron saturation current is also shown in figure 6. The intersection between this fit and the temperature fit occurs at the plasma potential. This results in a reading of Φ_{p} = -14.3 V.

## Calculating Electron Density

Knowing both the electron temperature and the ion saturation current allows us to calculate the electron density using,

where the new terms are M for the ion mass and A_{s} which represents the area of the probe sheath. For cases in which the applied probe bias does not greatly exceed the value needed to obtain one of the saturation currents we may approximate the sheath area as the probe tip area. For significant overbiasing this is not a good approximation. In most cases this criteria is met and the approximation is one of the smaller sources of error for probe measurements. The concepts of sheath expansion and Debye length with respect to probe size are worthy of discussion in another effort.

For argon we have M = 6.62 × 10^{−26} kg. The planar probe used to collect this data has an area of A_{probe} = 0.738 cm^{2} (this is the area of one side multiplied by two because it collects ions and electrons from both faces).

To calculate the electron density it is possible to directly insert the electron temperature in units of electron-Volts by noting the following relationship,

where on the left side the temperature is in units of Kelvin.

For the temperature measured in this setup, T_{e} = 3.70 eV, the electron density is found to be n_{e} = 8.09 × 10^{15} particles per cubic meter. Most plasma physicists use cgs units and would report this as a density of 8.09 × 10^{9} cm^{-3}.

## An Additional Concern for Temperature Analysis

This is a treatment to account for the presence of a hot electron distribution in the plasma. For plasmas that are generated through a fast electron breakdown process, such as an electron gun or cathode-anode pair, this effect is likely relevant.

In the preceding section the properties of our plasma have been determined under a few assumptions. Possibly the most significant assumption was that the plasma source (i.e. electron beam or electron spray) did not affect the IV trace. This is not absolutely correct and it is possible to include effects of the plasma source in our interpretation of the data. From a conceptual standpoint, we may expect that our system contains two separate electron populations. One population is the background plasma (hence referred to as the plasma electrons) and the other is from the plasma source directly (beam electrons). Since the plasma electrons are generated by ionization from collisions between beam electrons and background neutral gas they cannot possibly be more energetic than the beam electrons. This translates into the plasma electrons having less energy than the beam electrons, which is equivalent to them having a lower temperature, T_{e,plasma} < T_{e,beam}.

If there are two separate electron populations in the system, then it may further be expected that the IV trace displays two separate linear regions when plotted in the ln|I_{probe} – I_{sat}| fashion. In order to discern these regions it is necessary to better define the behavior that corresponds to the plasma temperature. If there is a hot electron tail (referred to as a tail because this represents the tail end of the electron velocity, or energy, distribution function), then those electrons will continue to strike the probe even after we have made V_{bias} more negative. If these electrons are affecting the probe measurement, then we should expect to see a second linear region in the logarithmic plot for values of V_{bias} approaching the floating potential. As previously, we must avoid actually considering the floating potential because that is where the I_{sat} effects have been subtracted to return a near zero electron current value.

Figure 7 shows a fit to the tail electron region of the curve. The slope of this line is the inverse of the hot electron temperature, T_{e,hot} = T_{e,beam}. The slope of this line (the blue line in figure 7) is less than that of the plasma temperature, but since the slope represent the inverse electron temperature this correctly gives a hotter temperature for these tail electrons. The fit gives T_{e,hot} ≈ 10 eV. The fit is made over the range −36 < V_{bias} < −22. A slightly different range may have provided a better fit. It is always a challenge to verify that you have made the best possible fit. An inclusion of error analysis and a quantitative measure of the quality of these linear fits will help justify your reported values.

It is not correct to use this temperature to calculate the density and plasma potential of the beam electrons. While they do skew the IV trace, we are still applying the general theory of quasi-neutrality and ion collection to the interpretation of the IV characteristic. There are no ions included in the beam distribution and the rest of our probe theory does not apply.

Conceptually, the claim is that the probe collects some of these beam electrons and treats them as plasma electrons. The probe cannot possibly know the difference between the different electrons it collects. To the extent that this simplified notion is true, the beam electrons result in an IV trace that returns a larger temperature than is correct for the plasma that the beam created.

## Sample Data for Practice

A spreadsheet file is available for download at the following link:

Sample Langmuir Data

As an actual data set this file contains noise and error, both of which contribute to the authentic experience of analysis. The values of plasma temperature, energetic tail temperature, floating potential, and ion saturation current that I obtained are included in the file for comparison to your own results. It is certainly possible to get a better (i.e. more accurate) result than I did.

If you are interested, then good luck.