To demonstrate controlling electron/metastable density ratio and electron temperature by applying negative voltages to the active (conducting) discharge wall in a low-pressure plasma with nonlocal electron energy distribution function, modeling has been performed in a short (lacking the positive-column region) direct-current glow discharge with a cold cathode. The applied negative voltage can modify the trapping of the low-energy part of the energetic electrons that are emitted from the cathode sheath and that arise from the atomic and molecular processes in the plasma within the device volume. These electrons are responsible for heating the slow, thermal electrons, while production of slow electrons (ions) and metastable atoms is mostly due to the energetic electrons with higher energies. Increasing electron temperature results in increasing decay rate of slow, thermal electrons (ions), while decay rate of metastable atoms and production rates of slow electrons (ions) and metastable atoms practically are unchanged. The result is in the variation of electron/metastable density ratio and electron temperature with the variation of the wall negative voltage.

Studies and better understanding of the short (lacking a positive-column region) direct-current (dc) discharges are interesting in many areas of science and technology including, but not limited to the examination of processes in cathode area of dc discharges,^{1} improvement of light sources,^{2} development of analytical gas sensors/detectors based on plasma-electron spectroscopy,^{3–5} and measurements of secondary electron emission in a plasma-solid boundary.^{6} It was demonstrated^{7} that the near-cathode region of a low-pressure dc discharge is characterized by an electron energy distribution function (EEDF) that is essentially nonlocal and that a description of this region should be generally two-dimensional.^{8} Consequently, effects attributed to the presence of energetic electrons in this region are well pronounced.^{9}

The influence of the discharge volume walls (the discharge tube radial boundary) on the plasma properties has been studied in Ref. 10. It was shown that one-dimensional models, like the von Engel–Steenbeck local model^{1,11} or the Kudryavtsev *et al*. nonlocal model,^{12} may miss transversal effects, for example, the possibility for the non-flat cathode sheath due to the variation of the current density over the cathode radius. Note that these effects have not been captured by the 2D-models in Refs. 7 and 8 because the effects are negligible for the investigated conditions reported therein. In Ref. 13, it was shown experimentally that the application of negative voltage to the discharge walls could increase the slow-electron temperature (which is typically is a fraction of 1 eV). However, the influence of the applied negative voltage on electron and metastable-atom density distributions in connection to the increasing temperature has not been studied. This is the goal of this paper.

In this study, the 2D model for the dc glow discharge in argon from Ref. 7 has been used for the characterization of the discharge device shown in Fig. 1 since this model adequately describes the details of two-dimensional effects such as the influence of walls on the discharge properties^{10} and the formation of a normal cathode spot on the cathode.^{7} The typical structure of the discharge plasma is shown in Fig. 1 and includes the cathode sheath, the negative glow (NG), the Faraday dark space (FDS), and the anode glow (AG). As this model has been described in detail elsewhere,^{7,14} only a short description will be given here.

In the discharge, shown in Fig. 1, ions from the plasma may reach the cathode surface with high energy (several hundreds of eV) and create secondary electrons launched back into the plasma with similar energy. Losing their energy in elastic and inelastic collisions with neutral atoms, these secondary electrons create slow, thermal electrons, ions, and metastable (excited) atoms. Thermal electrons have low energy (a fraction of eV). They are heated in collisions with energetic electrons.

The model is based on a fluid description of ions and neutral species (including ground and excited states) using the drift-diffusion approximation for the particle flux. The description of electrons is based on a “hybrid” approach, where electrons are separated into the bulk (slow, thermal) electrons and energetic electrons. The electron transport coefficients (the mobility *μ _{e}* and diffusion coefficient

*D*) are found from the solution of the Boltzmann equation using the standard two-term spherical harmonics expansion. The rates of electron impact reactions are calculated using the EEDF obtained from the solution of electron Boltzmann equation. The electron-energy balance equations for electron energy are also solved. This allows us to take into account the nonlocality of the electron kinetics over spatial distribution induced by electron thermal conductivity. The discretization scheme is based on the finite-volume approach. The Scharfetter-Gummel (exponential) scheme is used for the numerical discretization of the electron flux.

_{e}The plasma-chemical model may be described as follows. An effective excited state of the model includes metastable as well as resonance states of Ar. The set of reactions includes an elastic scattering of electrons, direct ionization of argon atoms, excitation of metastable argon states, stepwise ionization, Penning ionization of two argon metastable atoms, and excited-state irradiative de-excitation (including radiation trapping). The rates of processes have been calculated by convolving the EEDF, obtained from a solution of local Boltzmann kinetic equation, with the corresponding cross-sections. The cross-sections have been taken from Ref. 10.

The modeling has been performed in argon gas discharge with 1 Torr pressure. The length of the gas chamber was taken as 12 mm and radius was 12.5 mm. Wall was divided into three parts. The central part is metallic from 0.5 to 11.5 mm with a constant potential *V _{w}*. The side segments (near the cathode and the anode) were taken to be dielectric with dimensions of 0.5 mm. The value of electrostatic potential can be different at different points on the surfaces. This model corresponds to the experimental device shown in Fig. 1.

Three regimes have been modeled. In all three regimes, anode potential is equal to 0 V and cathode potential is −180 V. In regime “A,” wall potential is also equal to 0 V and therefore coincides to the anode potential. In regime “B,” wall potential is −25 V. In regime “C,” wall potential is −50 V. For the calculations, the coefficient of secondary emission of electrons from the cathode surface from ion bombarding has been taken to be independent of ion energy and equal to 0.1.

As an example of calculations, a typical result of 2D modeling of argon metastable density is shown in Fig. 2. In these figures, the top plot corresponds to regime “A,” middle plot corresponds to regime “B,” and bottom plot corresponds to regime “C.” To make modification of plasma properties more visible, Figs. 3–5 show the comparison of axial plasma properties for different regimes.

It is possible to see from Figs. 2 and 3 that by applying more negative voltage to the wall does not significantly change argon metastable atom density. In contrast, Fig. 4 shows that slow, thermal electron density (the same for ions) is significantly reduced by applying increasingly more negative potential to the walls. Fig. 5 shows considerable increasing electron temperature in the plasma volume with more negative wall voltage. Interpreted together, the ratio between slow, thermal electron (ion) density and metastable atom density decreases with increasing negative wall voltage and increasing slow-electron temperature.

An explanation of the observed phenomenon is as follows. Energetic electrons leave the cathode sheath with the energy of 180 eV (or below) and diffuse in the direction of the anode and walls while simultaneously ionizing and exciting metastable states of neutral atoms in the volume. During this process, they reduce their energy and create a continuous electron spectrum of energetic electrons at EEDF. Only the energetic electrons with energies *ε* > *eV _{w}* (where

*V*is the negative wall potential and

_{w}*e*is the electron charge) can reach the walls. Less energetic electrons can go to the anode only. Therefore, as the wall potential is essentially less than energy of the most of energetic electrons, the negative voltage

*V*, applied to the conducting (active) walls, can modify significantly the trapping of the low-energy part of energetic electrons (minority of the energetic electrons), while not essentially changing the high-energy (majority) part of those electrons. As a result, the excitation and ionization production depends only weakly on the applied potential to the wall. Metastable atoms disappear at the walls and during binary collisions (Penning ionization), independent of the wall potential. Therefore, the density of metastable atoms should not depend on negative wall potential, as confirmed in Figs. 2 and 3.

_{w}Similarly, the production of ions (slow, thermal electrons) depends weakly on the negative voltage applied to the walls. However, the heating of slow electrons is mostly due to the low-energy part of energetic electrons, as their collision frequency with slow electrons is nearly inversely proportional to $\epsilon 1.5$. At the same time, varying the negative wall potential modifies particularly the low-energy part of the energetic electrons and changes the heating the slow, thermal plasma electrons by the energetic electrons (as the heating is mostly due to the low-energy part of those electrons), which consequently changes the electron temperature and their diffusion rate to the anode. Therefore, due to the additional heating of the slow electrons (see Fig. 5), their diffusion to the anode will be faster and their density goes down with increasing negative voltage applied to the walls. As a result, while density of metastable atoms depends weakly on wall voltage, the density of slow electrons (ions) depends strongly on wall voltage, significantly reducing with increasing negative wall voltage. Figure 6 shows both behaviors more clearly. Note also that the electron current to the wall depends mostly on the low-energy part of energetic electrons. Calculations show that for regimes A, B, and C, the total current to the walls is −1.77 mA, −0.1 mA, and +0.02 mA (signs show that the first two currents are directed away from the wall and the last current is directed toward the wall).

Thus, the application of negative voltage to the discharge walls could change the trapping of the low-energy part of the energetic electrons that are emitted from the cathode sheath and that arise from the atomic and molecular processes in the plasma within the device volume. The low-energy part of the energetic electrons is responsible for heating the slow, thermal electrons. At the same time, the production of slow electrons and metastable atoms is mostly due to energetic electrons with higher energies. The variation of electron temperature results in a changing decay rate of slow, thermal electrons, while the decay rate of metastable atoms and production rates of slow electrons and metastable atoms are practically unchanged. The ability to control the electron/metastable density ratio and the electron temperature represents an important capability and is the main result, here.

The authors are grateful to Dr. I. D. Kaganovich for the valuable discussions. A part of this research was performed while one of the authors (V.I.D.) held a National Research Council Research Associateship Award at AFRL.