Electron Energy Distribution Function

Title: Exploring the Electron Energy Distribution Function: Unravelling Electron Behaviour in Plasmas 

Introduction 

Plasmas, the fourth state of matter, exhibit complex dynamics governed by the behaviour of their constituent particles. Understanding electron behaviour within plasmas is crucial for numerous applications, from plasma processing to fusion research and plasma chemistry. One key parameter that characterises electron behaviour is the electron energy distribution function (EEDF). Accurate measurement and characterisation of the EEDF provide valuable insights into electron energy, velocity, and flux. In this article, we will delve into the concept of the EEDF, its measurement techniques, factors influencing its shape, and the significance of precise EEDF measurement. We will also highlight Impedans Ltd’s diagnostic solutions in this field. 

Section 1: Understanding Electron Energy Distribution Function 

The electron energy distribution function refers to the probability density function that describes the energy distribution of electrons within a plasma. It provides valuable information about the population of electrons at different energy levels, velocities, and fluxes. By analysing the EEDF, researchers can extract key plasma parameters such as electron temperature (the width of the distribution, see article on electron temperature) and electron energy losses. 

The shape of the EEDF determines the behaviour of electrons in various processes within the plasma. Understanding the EEDF allows researchers to identify the dominant heating and cooling mechanisms in the plasma. For instance a high-energy tail in the EEDF indicates the presence of energetic electrons that contribute to plasma heating through collisional processes. Conversely, cooling mechanisms such as radiation may be evident in specific energy ranges of the EEDF.  

Another key dependency is that of chemical reactions within the plasma. The energy of electrons in the plasma determines their ability to participate in chemical reactions as different reactions have specific energy thresholds or activation energies. Formation and destruction rates of reactive species is also dependent on the shape of the EEDF which can influence the overall composition of the plasma and product distribution in chemical reactions. A measurement of EEDF is for researchers to accurately model the plasma chemistry to predict reaction rates, product concentrations and overall plasma behaviour.

The electron temperature can be extracted from the EEDF as given by the width of the energy distribution function. This can be quantified using the full width at half maximum (FWHM) or the most probable energy. Typically for low temperature plasmas, the EEDF is described using the Maxwell-Boltzmann distribution function. This provides a statistical description of how particle velocities, and in this case energies, are distributed within a plasma at thermal equilibrium. For electrons this distribution is given by:

Where f(v) is the electron number density function as a function of velocity, v, and T is the electron temperature (in energy units eV).

 Section 2: Factors Affecting the Electron Energy Distribution Function 

Several factors influence the shape and characteristics of the EEDF within a plasma. Electron density, electron temperature, and applied electric or magnetic fields play significant roles. 

Electron temperature affects the peak energy, the width and the high-energy tail of the EEDF. With higher temperatures shifting the distribution to higher energies, broadens the width of the EEDF, and makes the high-energy tail more pronounced. This indicates that the electron temperature raises the population of highly energetic electrons. The mathematical dependency of the EEDF on the electron temperature can be seen in the Maxwell-Boltzmann relation as given in section 1. 

Magnetic fields have a profound impact on the Electron Energy Distribution Function (EEDF) in a plasma. In plasma confinement devices, strong magnetic fields can cause particle drifts, leading to localized modifications of the EEDF. Magnetic mirrors can trap or reflect particles with different energies, altering the distribution’s shape and spread. Cyclotron motion induces energy exchange between particles, contributing to thermalization and potentially shaping the EEDF into a more Maxwellian form. Near surfaces, magnetic fields influence the plasma presheath and sheath regions, affecting particle energies and the EEDF. Furthermore, magnetic fields can induce electron heating and non-equilibrium behavior, leading to non-Maxwellian features like power-law tails. Understanding the interplay between magnetic fields and the EEDF is crucial for optimizing plasma behavior in magnetic confinement devices and other plasma applications

Section 3: Measurement Techniques for Electron Energy Distribution Function 

Several techniques have been developed for the accurate measurement of the EEDF. Common methods include Langmuir probes, Tomson scattering, and energy resolved mass spectrometry. 

Langmuir probes are widely used for EEDF measurement. They work based on the principle of the current-voltage characteristics of a biased electrode immersed in plasma. By analysing the collected current, researchers can derive information about the EEDF.

The EEDF may be found from the expression

The associated electron energy probability function EEPF is found from

Here, I and V represent the probe current measured at different applied voltages V.

Please refer to this article on Langmuir probes for more information on how they use this measured current to calculate fundamental plasma parameters. The advantages of this technique is that it gives a real time measurement of the EEDF, the set up is reasonably simple and cost effective and that it can be used for a wide range of plasma conditions and densities. Langmuir probes are the most common plasma diagnostic and have been used in research and industry for decades to understand plasma processes and behaviour. Howver this is an invasive technique as it requires placing the probe tip into the plasma, which may perturb it. The measurement is also sensitive to sheath effects near the probe tip. 

Another method is Tomson scattering, wherein a laser beam is directed into the plasma, and the scattered light is collected at various angles. The energy and frequency shifts in the scattered light provide information about electron temperature and density, from which the EEDF can be inferred. This is a non-intrusive, external technique which can provide spatially resolved measurements for high-temperature plasmas. However, it is limited to relatively high electron densities, and requires a complex set up of high-power lasers and sensitive detectors. 

Finally microwave diagnostics can also be used to measure the EEDF of a plasma. It operates in a similar technique to Tomson scattering however it uses microwaves signal instead of a laser beam and relies on the interaction of microwaves and electrons to change these signals when they are transmitted or reflected. Analysis of these changes allows for the determination of electron temperature and EEDF. This is also a non-invasive technique however it is very sensitive to magnetic fields present, and requires complex calibration and data interpretation. 

Section 4: Applications of Electron Energy Distribution Function Measurement 

The measurement of the Electron Energy Distribution Function (EEDF) finds diverse and crucial applications across various industries and scientific fields. In plasma processing, accurate EEDF measurements enable researchers and engineers to gain deeper insights into the energy distribution of electrons within the plasma. This knowledge is essential for precisely controlling plasma parameters, optimizing plasma processing techniques, and achieving improved material properties in applications like plasma etching, surface modification, and thin film deposition. 

 In fusion research, understanding the EEDF is of paramount importance for assessing plasma stability and confinement properties in magnetic confinement devices such as tokamaks. Measurements of the EEDF help scientists determine the energy content and distribution of charged particles, aiding in the design and optimization of fusion reactors. By studying the EEDF in these high-temperature plasma environments, researchers can identify factors influencing plasma behavior, plasma instabilities, and energy confinement, all crucial for the successful development of practical fusion power. 

In plasma chemistry, the EEDF plays a significant role in modeling and understanding chemical reactions within plasmas. By accurately measuring the EEDF, researchers can gain insights into the energy levels at which reactions occur, reaction rates, and species concentrations. The Lieberman global model, which incorporates the EEDF, is an example of how this distribution is utilized to predict reaction rates and species densities in plasma chemical processes. Additionally, sophisticated software packages like Bolsig+ leverage EEDF measurements to simulate and optimize electron-driven plasma chemistry, benefiting applications in microelectronics, pollution control, and materials processing. 

In low-pressure plasma applications, the measurement and understanding of the Electron Energy Distribution Function (EEDF) play a crucial role in optimizing plasma processes and enhancing device performance. The EEDF provides valuable information about the energy distribution of electrons, enabling researchers and engineers to tailor plasma conditions for specific applications. In low-pressure plasma systems, such as those used in plasma etching and thin-film deposition for microelectronics and semiconductor manufacturing, the EEDF influences the efficiency of chemical reactions and energy transfer processes. By characterizing the EEDF, researchers can precisely control electron energies, leading to improved material etching rates, surface selectivity, and device fabrication precision. Furthermore, in low-pressure plasma devices like capacitively coupled plasma (CCP) sources, understanding the EEDF is essential for optimizing plasma uniformity and stability, leading to more consistent and reliable plasma processing outcomes. 

Section 5: Impedans Ltd’s Solutions for Electron Energy Distribution Function Measurement 

Impedans Ltd offers advanced diagnostic solutions for precise and reliable EEDF measurement. Their Langmuir probe systems are designed to provide real-time data acquisition with high sensitivity. These systems offer compatibility with various plasma sources, allowing researchers and industrial users to gather accurate electron temperature measurements in different plasma environments. Below is an example of EEPF measurement taken from the software of one of their Langmuir probes in a typical Argon plasma. 

Impedans Ltd.’s diagnostic tools are user-friendly, making EEDF measurements accessible to researchers and engineers at all levels of expertise. A key feature is that all the complex Langmuir analysis is performed on the data automatically, with the calculations and methodology described in full to the user. This allows users to quickly and easily monitor plasma parameters such as the electron temperature, the EEDF, the plasma density and Debye length in real time. They also offer expertise training and guidance in using their plasma diagnostic devices through their customer support team, which allows the user to make the most of their plasma process. The company’s commitment to delivering accurate and reliable solutions has earned them a reputation as a trusted provider in the field of electron temperature measurement. 

 By leveraging Impedans Ltd.’s diagnostic solutions, researchers and industry professionals can gain valuable insights into their plasma processes. The obtained electron temperature data can be used to optimize process parameters, control plasma behaviour, and achieve desired outcomes in their specific applications. 

Conclusion 

The electron energy distribution function is a key parameter in plasma physics, offering valuable insights into electron behaviour, energy distribution, and flux within a plasma. Accurate measurement and characterisation of the EEDF are essential for optimising plasma processes, controlling plasma behaviour, and understanding electron-driven reactions. 

Impedans Ltd, with its expertise in diagnostic solutions, provides reliable and advanced tools for precise EEDF measurement. By leveraging Impedans Ltd‘’s cutting-edge technologies, researchers and industry professionals can unlock new frontiers in plasma physics and make significant advancements in their respective fields. For more information on these diagnostic solutions please see the Impedans website or get in contact with one of their expert engineers. 

References and further reading: 

• Lieberman, M. A., & Lichtenberg, A. J. (2005). Principles of Plasma Discharges and Materials Processing. John Wiley & Sons. 
• Chen, F. F. (1984). Introduction to Plasma Physics and Controlled Fusion (Vol. 1). Springer. 
• Phelps, A. V. (1994). Basic plasma processes on the electrode surfaces. Plasma Sources Science and Technology, 3(3), 431-440. 
• Czarnetzki, U., & Schulz-von der Gathen, V. (2012). Diagnostics of plasma-surface interactions: real-time measurements of the EEDF and energy flux to the surface. Plasma Sources Science and Technology, 21(3), 034005. 
• Guerra, V. (2002). Physics of the electron energy distribution function in low-pressure plasmas. Plasma Sources Science and Technology, 11(2), R1.