Improvements to EEDF Analysis from Langmuir Probes Using Integral Methods

Low density, low temperature plasmas are a vital area of interest in a broad range of applications ranging from nano-scale device fabrication to high efficiency lighting. Langmuir probes are considered a simple and cost efficient technique to diagnose plasmas, specifically, low temperature plasmas. By measuring the voltage/current relationship of the combined probe/plasma circuit, various plasma parameters can be determined. The most important plasma parameter measured by a Langmuir probe can be the Electron Energy Distribution Function (EEDF). Through the EEDF one can determine some characteristics of the plasma such as temperature and density.;In the 1940s Druyvesteyn published results stating that the EEDF can be obtained through the second derivative of the probe I--V characteristic through the electron transition portion of the probe trace. This method has been the most widely used for obtaining EEDFs. Since that time other techniques appeared later on in the 1970s taking advantage of Druyvesteyn's relation between the I--V characteristic and the EEDF is a Fredholm integral of the first kind, thus leading to the possibility of solving for the EEDF by Tikhonov regularization methods. These regularization techniques were rarely because there were challenges regarding the reconstruction of the EEDF.;In this work we are taking a closer look at both Druyvesteyn's relation and the Tikhonov method. For Tikhonov's method, we were able to overcome its challenges of over and under regularization, as well as the shift that occurs at the beginning of the distribution. This was carried out by the implementation of our proposed Hybrid method which acted directly on the singular eigenvalues of the kernel matrix, thus removing components that are highly corrupted by noise. The Druyvesteyn relation for EEDF's from Langmuir probes is derived based on a model that assumes spherical probe geometry. Most applications extend this formulation to arbitrary geometries including the more commonly used planar and cylindrical probes. In this work, we present a formulation of the relationship between electron current, probe potential, and EEDF for a cylindrical geometry that also accounts for conservation of angular momentum of the system and provides an identical integral relationship to that posed by Mott-Smith and Langmuir in 1926. This formulation shows a systematic defect in using the Druyvesteyn relation for cylindrical probes that becomes more pronounced for highly non-Maxwellian distributions. In order to minimize this geometry-induced distortion, solution of the integral relation between EEDF and probe current may be needed in place of the more commonly used derivative formulation of Druyvesteyn. Errors of the order 5--15% are common.