The main objective of the present paper is the detailed analysis of the electron kinetics near the Paschen's curve minimum of pulsed breakdown of nitrogen gas. Three main questions are to be answered. First, whether the breakdown curve minimum corresponds to the threshold electric field necessary for the runaway electron generation. Second, what is the role of these electrons in the vicinity of the Paschen's curve minimum. Third, what is the ionization cost near the minimum. To answer these questions, the one-dimensional model is used in which electrons are modeled using the Particle-in-Cell approach, while ions are modeled in the drift-diffusion approximation.

## I. INTRODUCTION

Today, pulsed discharges are used in many plasma applications. These include the dielectric barrier discharges for ozone generation,^{1} plasma assisted reforming^{2,3} and combustion,^{4} and plasma actuators.^{5} A sequence of nanosecond discharges is obtained every half period of dielectric barrier discharges. These discharges are also used for gaseous lasers, gaseous switches, and arc ignition.^{6,7} In the semiconductor industry, pulsed plasmas are used for the atomic layer etching.^{8}

Ignition of any gas discharge, no matter whether it is radio frequency or direct current discharge, is characterized by so-called breakdown or Paschen's curve.^{9} It is characterized by the left, i.e., the low-pressure, and the right, i.e., the high-pressure, branches. There is also the minimum of Paschen's curve. Both branches are characterized by different physical processes responsible for their formation. For instance, the left branch of direct current discharges involves surface processes such as the secondary electron emission due to the ion or metastable species impact or the photoelectron emission.^{10} The plasma of these discharges usually fills in the entire interelectrode volume, i.e., they develop in the so-called volumetric form. On the right branch, breakdown may develop either in the form of fast ionization waves or in the form of streamers.^{11,12}

Despite various breakdown mechanisms typical for both branches, there is one common feature characterizing them. This is the generation of so-called runaway electrons (RAEs), i.e., the electrons that gain more energy from the electric field along their mean free path than they lose in inelastic collisions.^{13–15} These electrons may be generated either from the seed background when the electric field within the cathode-anode gap exceeds the critical electric field, or from the secondary emitted electrons within the cathode sheath, or at the head of the fast ionization waves or streamers.^{16–19} The influence of these electrons on the formation of various forms of discharges has been established over the last two decades both experimentally and through numerical modeling (see Refs. 20–24 and references therein).

Most of the numerical modeling studies are devoted either to the left branch where many interesting surface phenomena are observed and explored^{9,10} or to the right branch.^{16,18} Indeed, physics at the right branch is probably one of the most studied fields of gas discharge physics over the last century. At the same time, very few studies were devoted to the thorough analysis of electron kinetics near the breakdown curve minimum especially for pulsed breakdowns.^{25–27}

Babich *et al.*^{26} have analyzed the Paschen's curves of pulsed breakdown in various gases. They assumed that on the left branch, gap breakdown is realized through the volumetric mechanism when all seeded electrons generate overlapping avalanches. On the right branch, it was assumed that the breakdown is realized through the streamer mechanism. Then, by applying the Raether's criterion on the streamer formation,^{28} they concluded that the minimum of the Paschen's curve corresponds to the critical electric field necessary for the RAE generation.

This observation complemented another important characteristic of the Paschen's curve minimum known for direct current breakdown since the 19th century.^{9} Namely, this point corresponds to the optimal conditions necessary for the electron multiplication. The ionization cost, i.e., the average energy spent on the production of one electron-ion pair, is maximal at the minimum of Paschen's curve and is called the Stoletov's constant. Its value for pulsed nanosecond discharges was calculated by Macheret *et al.*^{29,30} Note that in Refs. 29 and 30, the ionization cost necessary for the sustaining periodic pulsed discharge was calculated, i.e., the secondary electron emission from the cathode was accounted for. In the present paper, the focus is only on the breakdown stage, i.e., on the stage of dense plasma generation. The ions of this plasma induce secondary electron emission from the cathode on much longer timescale, which ignites the self-sustaining pulsed discharge.

The goal of the present paper is the analysis of the electron kinetics near the minimum of breakdown curves of nanosecond discharges in molecular nitrogen. This gas is used for the growth of nitrides by plasma assisted beam epitaxy,^{31} for nitridation processes,^{32} to etch organic films with low dielectric constant,^{33} etc.

There are three main objectives of this analysis, which are the understanding of the role of runaway electrons during breakdown, checking whether the ionization cost is minimal at the minimum of Paschen's curve of nanosecond discharge, and checking whether the Paschen's curve minimum corresponds to the critical electric field necessary for the runaway electrons' generation. This paper is organized as follows. Section II briefly overviews the numerical model used in the present studies. Section III presents the results and discusses the electron kinetics around the minimum of the breakdown curves. Finally, Sec. IV summarizes the results of the present studies.

## II. BRIEF OVERVIEW OF THE NUMERICAL MODEL

The model used in the present studies is the one-dimensional kinetic model previously applied to analyze pulsed breakdown at various conditions.^{34,35} In this model, the electrons are described using the particle-in-cell Monte Carlo collisions method, while the ions are modeled in the drift-diffusion approximation. The working gas is nitrogen (N_{2}) whose density is kept constant along the cathode-anode gap. The gas temperature is 300 K. In the present model, several types of collisions between the electrons and N_{2} were considered. These were the electron-neutral momentum transfer, the electron energy losses to the excitation of vibrational and electronic levels of N_{2}, and the dissociation of N_{2}. These collisions did not result in the production of new particles but were accounted for the electron energy losses only. Also, the ionization of N_{2} was considered, which resulted in the generation of a new electron–ion (N_{2}^{+}) pair. The cross sections of these processes were taken from Itikawa's database.^{36} The transport properties of N_{2}^{+} ions were taken from Ref. 37.

The cathode–anode gap was $d=$ 1 cm, and the right boundary (anode) was kept grounded. The left boundary (cathode) potential was rising with time as $\phi C(t)=\phi 0\tau t$, where $\phi 0=$ −100 kV and $\tau $ is the voltage rise time. Three values of $\tau $ were considered in the present paper: 100, 10, and 1 ns. Initially, the cathode-anode gap was seeded with plasma having the density of 10^{10} m^{−3}. The time step was 10^{−14} s in all simulations, and the numerical cell size was 5 *μ*m.

For the rise times considered in this paper, the gap breaks down on the nanosecond timescale. The ion transit time through the gap is much longer than the breakdown time. Therefore, they can be considered motionless, and the secondary electron emission from the cathode can be neglected.

## III. RESULTS AND DISCUSSION

### A. Breakdown voltage as the function of gas pressure and voltage rise time

This subsection discusses the influence of the gas pressure and the voltage rise time on the breakdown curves. Here, the gap breakdown voltage is defined as the cathode potential at which the space charge of plasma generated within the gap starts shielding the applied electric field. Depending on the conditions, this is obtained for the plasma density $ne\u223c$ 10^{16}–10^{18} m^{−3}, namely, the faster the rise time, the larger the density at which breakdown is obtained.

The typical dynamics of gap breakdown from homogeneously seeded electrons was presented, for instance, in Ref. 38. Figure 1 shows the time evolution of the electric field in the center of the simulation domain for the voltage rise time of 1 ns and the gas pressure of 9 kPa. One can see that at *t* < 0.3 ns, the electric field rises linearly with time. Here, the plasma density is such that it does not affect the applied electric field. However, at *t* ∼ 0.3 ns, the electric field starts decreasing, which, according to the aforementioned definition, means the electrical breakdown of the gap.

The breakdown voltage as the function of gas pressure for three different voltage rise times is shown in Fig. 2. These results agree with the experimentally obtained data^{26} and the results of numerical modeling studies using kinetic approach.^{27} This figure also shows the breakdown voltage obtained in the case when the gas ionization by electrons having the energy larger than $\epsilon th=$ 100 eV is forbidden. The energy losses of these energetic electrons decrease for increasing electron energy. Therefore, they are usually considered as RAE in the literature.^{25} In these simulations, the electron with $\epsilon e>\epsilon th$ loses the energy equal to the ionization threshold of N_{2} but does not produce a new electron–ion pair.

One can see from Fig. 2 that all three curves have the minima typical for Paschen's curves. They have the low- and high-pressure branches and the minimum. The decrease in the voltage rise time shifts the breakdown curve minimum toward higher pressures and higher voltages, i.e., the breakdown voltage for any given gas is not the unique function of $Pd$, where $P$ is the gas pressure, but depends on the voltage rise time.^{26,27}

Figure 2 shows that the model that accounts for the gas ionization by only the electrons with $\epsilon e<\epsilon th$ also predicts the minima of the breakdown curves, i.e., the presence of runaway electrons is not crucial for the observation of two branches. This also means that the participation of RAE in volumetric breakdown in the vicinity of the Paschen's curve minimum is not crucial. This result does not support the conclusion of Babich *et al.*^{25,26,39} about their importance for the formation of the left branch. Our simulation results have shown that the neglection of the gas ionization by high-energy electrons shifts the left branch of the breakdown curves toward higher pressures. At high pressures, when the electron mean free path becomes short (≪1 mm), and the RAE formation is not observed, the curves with and without threshold coincide. This effect is explained in Sec. III B by analyzing the contribution of different electron groups to the gas breakdown.

### B. Nature of the breakdown curve minimum for the voltage rise time of 1 ns

^{40}In the tail of the EEDF, this assumption may be inaccurate. Then, the Townsend ionization coefficient can be calculated by

Figure 3 shows that these coefficients depend not only on the reduced electric field $E/ng$ but also on the voltage rise time. As was discussed in Ref. 27, this is obtained due to the spatial and temporal non-locality of the EEDF.

Figure 4 shows the EEDF for three values of the gas pressure for the voltage rise time of 1 ns. The inset in Fig. 4(a) shows the anisotropic electron velocity distribution function (EVDF) obtained along the electric field. Similar EVDFs were obtained for 9 and 20 kPa. One can see that the EEDF can be fitted by the bi-Maxwellian distribution functions, which are separated by the energy ∼100 eV. This energy corresponds to the value at which the ionization cross section of N_{2} obtains the highest value. The comparison between these three figures shows that the highest energy of electrons obtained in the cathode-anode gap decreases for increasing gas pressure. For the gas pressure of 4 kPa, the highest energy is ∼2–5 keV; for the pressure of 9 kPa, it is ∼1 keV; and for the pressure of 20 keV, it is ∼400 eV. These tail electrons obtain more energy from the electric field along their mean free path than they lose in inelastic collisions with N_{2}. Therefore, they can be considered as runaway.

The generation of these electrons represents the spatial non-locality of the EEDF. Note that the generation of RAE is observed not only at the left branch of the Paschen's curve but also at the right branch. This means that for the voltage rise time of 1 ns, the breakdown curve minimum does not correspond to the threshold electric field necessary for the RAE generation as was pointed out in Ref. 25. Our simulation results show that for the fast rise times, they are generated even at the right branch although their peak energy and density are much smaller than those observed near the breakdown curve minimum and at the left branch.

The temporal non-locality of the EEDF is observed on the left branch of the breakdown curve. The ionization rate coefficient is defined by the part of the EEDF, where $\epsilon e>\epsilon ion$ [see Eq. (1)], while the electron mobility is defined by the entire EEDF [see Eq. (2)]. Our simulation results have shown that for any given $E/ng$, on the left branch, $kion$ reaches the steady state on the sub-nanosecond timescale, while $\mu e$ reaches the steady state much slower, on the breakdown timescale. This also means that the Townsend ionization coefficient is the non-local in space and time at the left branch of the breakdown curve.

Figure 5 shows the normalized number of each type of electron-neutral collisions as the function of time for three values of the background gas pressure. One can see some general trends for all values of the gas pressure. Namely, initially, when the electric field between the electrodes is small and the electron average energy is also small, the dominant channel of electron energy losses is the excitation of vibrational levels of N_{2}. These levels have low energy thresholds (∼1–2 eV) and large cross sections. Therefore, the electrons present in the gap mainly lose their energy to excite these levels.

At later times, when the average electron energy increases due to the increasing electric field, the dominant collisions become the ionization and the dissociation of the background gas. One can see that for the gas pressure of 4 kPa, the electron energy spent to the excitation of vibrational and electronic levels of N_{2} is less than 1%. The main part of energy is spent to the gas ionization (∼70%) and to the dissociation (∼30%). One can see that the increase in the gas pressure results in the decrease in the fraction of energy spent on the ionization, while the fraction of energy spent on the dissociation remains almost constant. For the gas pressure of 9 kPa, ∼62% is spent to the ionization and ∼30% is spent to the dissociation, while for the gas pressure of 20 kPa, ∼58% is spent to the ionization and ∼32% is spent to the dissociation.

The ionization cost for three values of the gas pressure as the function of time is shown in Fig. 6(a). Here, the ionization cost was calculated as the ratio between the energy gained by all electrons presented in the simulation domain from the electric field to the total energy spent by these electrons on the gas ionization. One can see that initially, the ionization cost is extremely high. This is obtained due to the small electric field between the electrodes, which accelerates very few electrons to the energies exceeding the ionization threshold. Since there are many electrons within the gap and all of them are accelerated by the electric field, the ratio between the energy gained by these electrons and only a few ionization events result in large ionization cost.

The increase in the electric field results in the increase in the average electron energy and in the increase in the number of electrons with $\epsilon e>\epsilon ion$. This results in a decrease in the ionization cost. Interestingly, for the gas pressure of 4 kPa, the ionization cost obtains the minimum value ∼500 eV at *t* ∼ 0.8 ns and then starts increasing again. To understand this effect, Fig. 7 shows the normalized number of ionizations by four different electron groups, which are defined by their energy. Group 1 has energy below 100 eV, and the ionization cross section for this group is the increasing function of the electron energy. Group 2 has the energy in the range 100 eV $<\epsilon e<$ 200 eV, group 3 has the energy in the range 200 eV $<\epsilon e<$ 1 keV, and group 4 has the energy exceeding 1 keV. The electron energy losses in inelastic collisions of groups 2–4 decrease for increasing $\epsilon e$.

One can see from Fig. 7(a) that for the gas pressure of 4 kPa, first three groups contribute almost equally to the gas ionization at *t* > 0.1 ns. Moreover, at *t* > 0.23 ns, group 3 generates up to 40% of the electron–ion pairs. The increasing contribution of the high energy electrons to the gas ionization means that more energy is spent to accelerate the electron to such an energy that it can experience one ionizing collision. The latter is a rare event due to the long mean free paths of keV electrons. Also note that the ionization cross section of the group 3 electrons is smaller than that of groups 1 and 2.

The increase in the gas pressure leads to a significant decrease in the ionization cost. Here, it is important to note that the smallest ionization cost is obtained for the gas pressure of 20 kPa (∼110 eV) and not for the pressure of 4 kPa (∼220 eV), i.e., at the right branch of the breakdown curve and not at its minimum, which does not support the conclusions of Ref. 25. This result can also be understood from Figs. 7(b) and 7(c), which show that for 20 kPa, more than 80% of the ionizing collisions are by the electrons with $\epsilon e<$ 100 eV. That is, the electron obtains smaller energy from the electric field before experiencing the ionizing collision.

The electron density is defined by both the electron drift in the applied electric field and by the ionization rate coefficient. Figure 3(a) shows that the increase in the gas pressure results in the increase in $kion$ for any given $E/ng$. However, this increase is more drastic when the pressure increases from 4 to 9 kPa, than when it increases from 9 to 20 kPa. Figure 3(b) shows that the increase in the gas pressure results in the decrease in the electron mobility. Thus, the increase in the gas pressure results in the increase in the electron density. The multiplication of decreasing $\mu e$ and increasing $ne$ for increasing pressure in Eq. (4) results in the maximum of the plasma conductivity for the gas pressure of 9 kPa.

### C. Influence of the voltage rise time on electron kinetics near the breakdown curve minimum

In this subsection, the influence of the voltage rise time on the electron kinetics near the breakdown curves minima is investigated. For the rise time of 1 ns, the breakdown minimum is obtained for the gas pressure ∼9 kPa (see discussion in Sec. III B); for $\tau =$ 10 ns, it is ∼2.2 kPa; and for $\tau =$ 100 ns, it is ∼1.4 kPa.

Figure 8 shows the ionization rate coefficient and the electron mobility as the functions of reduced electric field for three values of the voltage rise time. One can see that both not only are the non-unique functions of $E/ng$ but also depend on $\tau $, which is explained by the spatial and temporal non-locality of the EEDF (see Sec. III B). Figure 8(a) shows that the ionization rate coefficient for all three $\tau $ converges to one function for high $E/ng$, while Fig. 8(b) shows that $\mu e$ increases for increasing $\tau $.

The normalized number of different electron-N_{2} inelastic collisions is shown in Fig. 9. The trends are like those discussed in Sec. III B, namely, the dominant electron-neutral collision during the breakdown is the ionization. One can see that the faster the rise time, the larger the number of ionizing collisions and the smaller the number of the dissociating collisions. However, the difference is <5%. Therefore, one can conclude that the voltage rise time has insignificant influence on how the electrons spend their energy.

The contribution of different electron groups to the ionization is shown in Fig. 10. One can see that for $\tau =$ 100 ns, more than 80% of the ionizations is by the group 1 electrons and ∼20% is by the group 2 electrons. The contribution of groups 2 and 3 is negligibly small. For faster rise times, the contribution of group 1 decreases, while the contribution of groups 2 and 3 increases. One can see that for the rise times of 10 and 1 ns, the group 2 electrons contribute up to ∼30%, while the contribution of group 1 electrons drops to ∼50%–60%. The remaining ∼10%–20% is by the group 3 electrons. These results may be explained by much larger breakdown voltage for $\tau =$ 1 ns than for $\tau =$ 100 ns. Indeed, the breakdown voltage for $\tau =$ 100 ns is ∼2.4 kV, while for $\tau =$ 1 ns, it is ∼28.8 kV. This means that in the former case, the electrons pass through smaller potential difference. Therefore, their average and peak energies are smaller for $\tau =$ 100 ns. Also, the energy gained by the electrons from the electric field along their mean free path is ∼2 times larger for shorter rise time.

Such contribution of different electron groups to the gas ionization influences on the ionization cost, which is shown in Fig. 11. One can see that the longer the rise time, the larger the ionization cost. For $\tau =$ 100 ns, the ionization cost minimum is ∼1.5 keV, while for $\tau =$ 1 ns, it is ∼200 eV. Such dependence can be explained as follows.

The simulation results show that the longer the rise time, the smaller the breakdown electric field and the smaller the gas pressure at which the breakdown curve minimum is observed. The smaller breakdown field means that the gap breakdown is obtained at smaller value of the electron density within the gap. Figure 8(a) shows that we can assume that the breakdown is obtained for such $E/ng$ that the ionization rate coefficient is constant. Then, the number of ionizations is proportional to the gas pressure. It was obtained that the breakdown voltage decreases faster for increasing rise time than the gas pressure decreases. Therefore, the slower the rise time, the larger the ionization cost.

## IV. SUMMARY

In the present paper, the one-dimensional kinetic model was used to analyze the electron kinetics in the vicinity of the breakdown curves minima of pulsed discharges in molecular nitrogen. In this model, only the gas phase processes were considered such as the electron-neutral collisions, while the secondary electron emission from the walls was neglected due to the fast voltage rise times.

The simulation results have shown that the contribution of runaway electrons to the gas ionization for a fixed value of the voltage rise time decreases when moving from the left branch to the right. The ionization cost also decreases for increasing gas pressure. It was obtained that the increase in the voltage rise time results in the increase in the ionization cost obtained in the breakdown curve minimum.

Three main results were obtained in the present studies. First, the breakdown curve minima do not correspond to the threshold electric field necessary for the runaway electrons' generation. These electrons were observed at both the left and the right branches of the breakdown curve. Second, the gas ionization by runaway electrons is not crucial for the observation of the breakdown minimum although their contribution to the ionization process influences the location of this minimum. The third result obtained here is that the ionization cost is not minimal at the breakdown curve minimum. The optimal ionization cost is obtained at the right branch of the breakdown curve.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Dmitry Levko:** Conceptualization (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Physical Chemistry of Ozone*

*Plasma Chemistry*

*Pulsed Gas Laser*

*Physics of Spark Discharge*

*Gas Discharge Physics*

*The Physics of Pulse Breakdown*

*High-Energy Phenomena in Electric Discharges in Dense Gases: Theory, Experiment, and Natural Phenomena (ISTC Science and Technology Series, Vol. 2)*

*Electron Avalanches and Breakdown in Gases*

*43rd AIAA Aerospace Sciences Meeting and Exhibit*

*Fundamentals of Plasma Physics*