Electron kinetics in atmospheric-pressure argon and nitrogen microwave (4 GHz) microdischarges is studied using a self-consistent one-dimensional Particle-in-Cell Monte Carlo Collisions model. The reversal of electric field (i.e., inverted sheath formation) is obtained in nitrogen and is not obtained in argon. This is explained by the different energy dependencies of electron-neutral collision cross sections in atomic and molecular gases and, as a consequence, different drag force acting on electrons. A non-local behavior of electron energy distribution function is obtained in both gases owing to electrons are generated in the plasma sheath. In both gases, electron energy relaxation length is comparable with the interelectrode gap, and therefore, they penetrate the plasma bulk with large energies.

## I. INTRODUCTION

Generation of stable high-density non-equilibrium atmospheric pressure plasmas at small length scales has many challenges. These include instability with respect to the glow-to-arc transitions, damage of electrodes by energetic ion bombardment and by chemically reactive species. One of the few possible approaches to avoid these drawbacks is the use of microwave (mw) generated microdischarges.^{1,2} Microdischarges are discharges whose characteristic length is less than 1 mm in at least one of the geometric dimensions. The interest in these discharges has grown in the last decade due to prospective applications in electric propulsion,^{3} plasma medicine,^{4} material processing,^{5} chemical synthesis,^{5} and metamaterials for the manipulation of electromagnetic waves.^{6}

The use of microwaves (mw) (*ν* > 1 GHz) for the generation of microplasmas has many advantages over the use of direct-current or low-frequency excitation. Principally, electrons and ions can be confined in the interelectrode gap by oscillating electric field with low effective plasma potential, which results in low particle losses to the walls. This leads to an increase in the plasma density because each electron resides in plasma for a long time and can generate many electron–ion pairs. Also, the decrease in the ion wall flux and the low ion impact energy at the electrode owing to the low plasma potential minimize heating and damage of electrodes.

Recent progress in the understanding of microwave microdischarges through both experimental and numerical simulation studies can be found in Refs. 7–15. The importance of numerical simulations cannot be overstated since they allow obtaining physical insights under conditions where experimental study is difficult.

In the present paper, we continue our study of atmospheric-pressure mw microdischarges reported earlier.^{14,15} Here, we present the results of the study of electron kinetics in these microdischarges. This question was addressed previously in Ref. 8, where the authors studied the electron kinetics in radio-frequency (13.56 MHz) atmospheric-pressure helium microdischarge (75, 100, and 200 *μ*m gaps). Their excitation frequency was clearly far from the mw range. Our simulations are carried out using self-consistent one-dimensional Particle-in-Cell Monte Carlo Collisions (1D PIC/MCC) model. Microdischarges in molecular nitrogen (N_{2}) and argon (Ar) are studied. The understanding of processes in these two gases is of principal importance. Nitrogen is the main component of air which is used in open discharges, while argon is the noble gas (i.e., chemically inactive) and is widely used as the working gas in fundamental discharge studies.

## II. THEORETICAL BACKGROUND

Let us discuss the parameters that allow us to distinguish different regimes of alternating current (ac) discharges.^{16} Electrons in plasma are described by the distribution function *f*(*x*,*v*,*t*), which is the function of three spatial coordinates and three components of velocity and time. This function is determined by solving the Boltzmann kinetic equation^{16}

Here, *q _{e}* is the elementary charge,

*m*is the electron mass,

_{e}*E*is the electric field, and

*St*(

*f*) is the collision integral.

In non-equilibrium plasmas, the standard method to solve Eq. (1) takes advantage of the relatively small directional anisotropy of electron velocity distribution function. For atmospheric-pressure discharges, this is usually true due to frequent elastic electron-neutral collisions which can drastically change the direction of electron velocity but does not significantly influence the magnitude of velocity owing to the small ratio between electron and neutral (*M*) masses. Therefore, electron motion in the electric field is diffusion-like. Thus, the function *f*(*x*,*v*,*t*) can be expanded as^{16}

where $f0$ and $f1$ are the isotropic and anisotropic parts of electron velocity distribution function, respectively, and $\u03d1$ is the angle between the direction of electron propagation and electric field. Function $f0$ is usually used in fluid models in order to obtain electron transport coefficients and rate coefficients of electron-neutral reactions.

Equations for $f0$ and $f1$ can be found, for instance, in Ref. 16. Here, we discuss only the conditions for which Eq. (2) can be applied. It was shown in Ref. 16 that when $\omega \nu m\u226a1$ (*ω* is the mw angular frequency and $\nu m$ is the momentum transfer collision frequency), function $f1$ can be considered as stationary. At atmospheric pressure, $\nu m\u2009$ is ∼10^{12} s^{−1}. Hence, for 4 GHz condition, $\omega \nu m\u226a1$ is satisfied and the approximation (2) is valid if the following conditions are satisfied:

Here, $\nu *$ is the frequency of inelastic electron-neutral collisions, $\lambda $ is the electron mean free path, $\epsilon e$ is the electron energy, and $\Lambda $ is the characteristic diffusion length. In 1D plane geometry, the latter is defined as $\Lambda =dgap/\pi $, where $dgap$ is the interelectrode gap.

For atmospheric pressure, condition (3a) is always satisfied. Condition (3b) means that the energy gain of an electron from the electric field during its propagation along a mean free path is much smaller than the average electron energy.^{16} This also means that the electron drift velocity is much smaller than the electron thermal velocity. Condition (3c) is usually satisfied at the atmospheric pressure. Thus, if (3a)–(3c) are satisfied, the electron distribution function is defined by the local value of the electric field.^{16}

The isotropic part $f0$ defines the electron energy distribution function (EEDF). This function is formed with a typical time scale $\tau \epsilon $ which is the energy relaxation time due to elastic and inelastic collisions. In the case of dominant elastic collisions, $\tau \epsilon =1/(\delta \u22c5\nu m)$, where $\delta =2me/M$. In the case of dominant inelastic collisions, $\tau \epsilon =1/\nu *$. Then, one can classify ac discharges as follows:^{16}

- If $\omega \u22c5\tau \epsilon $ ≫ 1, electric field varies faster than the energy relaxation occurs. Then, $f0$ does not depend explicitly on the time varying electric field but is defined by an effective electric field(4)$Eeff=E\u03032\nu m\nu m2+\omega 2.$
Here, $E\u0303$ is the average electric field.

If $\omega \u22c5\tau \epsilon $ ≪ 1, the energy relaxes faster than the time variation of electric field, i.e., $f0$ follows the time evolution of the applied electric field.

Cross sections of electron-neutral collisions are the functions of electron energy. Therefore, in mw fields, one can obtain a situation where the body of the EEDF is defined by the effective electric field but its tail is defined by the instantaneous electric field (see Sec. III).

Formation of EEDF is also characterized by the length scale $\lambda \epsilon $, which is the energy relaxation length. If the dominant collisions are the momentum transfer collisions, it is defined as

If the inelastic collisions play a significant role in the energy balance, one has

If $\lambda \epsilon \u226a\Lambda $, $f0$ does not depend on the spatial gradients and is defined by the local value of the electric field. Otherwise, $f0$ does depend on the gradients. This is usually obtained if (3b) is not valid.

At the atmospheric pressure conditions, the electron mean free path is estimated as $\lambda =1ng\sigma \u223c4\u2009\mu m$. Here, *n _{g}* is the background gas density and $\sigma \u223c10\u221216\u2009cm2$ is the electron-neutral momentum transfer collision cross section which we took for low-energy electrons (see Fig. 1). In nitrogen, one has $\delta \u22480.006$. Thus, $\lambda \epsilon \u223c64\u2009\mu m$, i.e., for mw microdischarge, the energy relaxation length can be comparable with the interelectrode gap.

## III. RESULTS AND DISCUSSION

The 1D PIC/MCC model used in this study was detailed in our previous papers.^{14,15} This model includes electron-neutral collisions, electron secondary emission from both electrodes induced by electrons and ions, and elastic electron reflection. Two gases are considered. The first gas is N_{2} for which we consider elastic and inelastic (ionization, excitation of vibrational and electronic states, and dissociation) electron-neutral collisions. The second gas is Ar for which we consider elastic collisions, ionization, and excitation of the first electronic level. For convenience, the cross sections of these processes are shown in Fig. 1.^{17}

Both gases are at the atmospheric pressure and room temperature. The interelectrode gap is $dgap$ = 60 *μ*m, and the mw frequency is *ν* = 4 GHz. The voltage $U(t)=Ubr\u2009sin(2\pi \nu t)$ ($\omega =2\pi \nu $) is applied to the left electrode, while the right electrode is grounded. Here, $Ubr$ is the breakdown voltage for the given excitation frequency. Our simulation results have shown that for N_{2}, $Ubr$ = 190 V, and for Ar, $Ubr$ = 100 V. Therefore, simulations were carried out for these two values of $Ubr$.

The interelectrode gap is divided into 1000 equal intervals (i.e., space step is $\Delta x$ = 0.06 *μ*m). The choice of time step is dictated by the Courant condition, i.e., $\Delta t=\Delta x/(2vmax)$, where $vmax$ is the largest electron velocity obtained in simulations. At these conditions, $\Delta t$ ≈ 4 × 10^{−15} s. The number of electron and ion macro-particles used in our simulations was ∼ 5 × 10^{5} each.

### A. Discharge in molecular nitrogen

Figures 2–6 show the results obtained for N_{2}. One can see the generation of dense plasma in the interelectrode gap whose peak density is $ne$ ∼ 2 × 10^{20} m^{−3} (see Fig. 2(a)). This plasma screens the external electric field leading to its concentration mainly in the plasma sheaths near both electrodes (see Fig. 2(b)). Electron average energy obtained in simulations is ∼0.6 eV, while the ion temperature remains equal to the background gas temperature (0.026 eV).

The electron heating profile is shown in Fig. 2(c). In our previous paper,^{15} we showed that the electron stochastic heating due to the sheaths oscillations is an inefficient mechanism for electron heating. At the considered conditions, the dominant mechanism is the Ohmic (Joule) heating. One can see two peaks of power at the same instant of time. The first peak corresponds to the expanding sheath which accelerates electrons that reached this electrode during preceding sheath collapse. The electron mean free path is ∼2 *μ*m, which is much smaller than the sheath thickness (∼10 *μ*m). Therefore, sheath is collisional and electrons dissipate part of their energy during their propagation throughout the sheath. Collisions limit the largest value of the electron energy obtained to be ∼25 eV. The second peak occurs near the opposite electrode, which means the reversal of electric field at that location (Fig. 2(b)). In this phenomenon, plasma sheath near the instantaneous grounded electrode accelerates electrons toward this electrode instead of their trapping in the plasma bulk which is obtained in conventional discharges. Also, it is important to note that the maximum values of the electron heating in both peaks are comparable in spite of much larger electric field in the expanding sheath. This is explained by much larger electron density in the region of reversed electric field.

Electric field reversal (inverted sheath) was previously obtained using 1D PIC/MCC simulations in the low-pressure (40 Pa) radio-frequency (13.56 MHz) discharge in N_{2}. The interelectrode gap was 4 cm. This effect is attributed to the collisional drag force on electrons moving into the sheath.^{18} This means that electrons do not respond instantly to the variations of electric field. The authors in Ref. 18 derived an expression for the maximum value of the reversed field *E _{R}*

Here, $sm$ is the sum of two sheaths widths. Substituting $sm$ ≈ 20 *μ*m (Fig. 2(a)) and $\nu m$ ≈ 5 × 10^{12} s^{−1}, we find $ER$ ≈ 7 × 10^{6} V/m. This is in good agreement with our simulation results. Thus, it may be argued that at atmospheric pressure, electric field reversal is also caused by the electron drag force.

The reversal of electric field influences significantly the production of energetic electrons (Fig. 3). One can see from Fig. 3(a) that the main part of electrons have energy less than 5 eV. Figure 1 shows that in N_{2} gas, these electrons can only experience momentum transfer collisions and excite vibrational states of N_{2} (Figs. 4(a) and 5(a), respectively).

Electrons having energy in the range 5 eV ≤ *ε _{e}* ≤ 15 eV are obtained in the entire interelectrode gap (Fig. 3(b)). However, the peak values of their density are obtained near the electrodes in the locations of the peak values of electron heating (Fig. 2(c)). These electrons can efficiently dissociate molecules N

_{2}and excite their electronic states (Figs. 4(c) and 5(b), respectively). The cross section of vibrational excitation at

*ε*> 5 eV is small (Fig. 1(b)). Thus, electrons from the second group cannot produce vibrationally excited N

_{e}_{2}.

The most energetic electrons (*ε _{e}* > 15 eV) are obtained only near the electrodes (Fig. 3(c)), where the Ohmic heating is the most efficient (Fig. 2(c)). Only these electrons are able to ionize N

_{2}, thus maintaining the plasma.

The comparison between Figs. 2(c) and 5(a) shows that the largest rate of vibrationally excited states of N_{2} production is obtained in the region of reversed electric field. This is due to the large electron density (Fig. 2(a)) and favorable average electron energy in that location. Namely, our simulation results have shown the average electron energy in the sheath ∼5 eV while in the reversed field ∼1 eV. Figure 1(b) shows that the cross section of vibrational excitation of N_{2} has the largest value for electrons with *ε _{e}* ∼ 1 eV and decreases for increasing

*ε*. Therefore, the rate of vibrational states production of N

_{e}_{2}is the largest in the region of reversed electric field, where

*ε*∼ 1 eV.

_{e}The dissipation of electron energy in different collisions is shown in Figs. 4 and 5. As expected, the dominant collision is the elastic scattering, because in N_{2} gas, momentum transfer cross section has the largest value among all collision cross sections for *ε _{e}* < 100 eV (Fig. 1). The dominant inelastic process is the excitation of vibrational states of N

_{2}. Because the energy threshold of this process is ∼0.3 eV, electrons experience these collisions in the entire interelectrode gap. Other inelastic processes have significantly larger energy thresholds (Fig. 1). Therefore, electrons ionize, dissociate, and excite electronic states only near the electrodes where the largest electron energy is obtained.

One can conclude from Figs. 4 and 5 that in N_{2} gas, only a small fraction of input mw power goes to the plasma generation. Most of the energy goes to the increase of the chemical reactivity of plasma such as production of reactive nitrogen atoms and excited states.

Time evolution of the electron energy distribution function (EEDF) at three different locations of the interelectrode gap is shown in Fig. 6. In order to understand its evolution, we plot in Fig. 7 the ratios $\omega /\nu $ and $\lambda \epsilon /\Lambda $ (see Sec. II). Figure 6 shows that the EEDFs can be divided into two main regions, namely, the low-energy body (plasma electrons) and high-energy tail.

Figure 7 allows us to understand different parts of EEDFs seen in Fig. 6. Figure 7(a) shows that for $\epsilon e$ ≈ 1.5 eV and $\epsilon e$ ≈ 4 eV, vibrational excitation of N_{2} is non-negligible and $\omega /\nu vibr$ ∼ 1. In the region 1.5 eV < $\epsilon e$ < 4 eV, the cross section of vibrational excitation is zero (Fig. 1(b)). Therefore, there is no vibrational excitation in this energy range ($\omega /\nu vibr$ is put zero in Fig. 7(a)). Also, for $\epsilon e$ > 11 eV, inelastic energy losses to electronic excitation and ionization become important and one obtains $\omega /\nu $ ≪ 1. Thus, the energy relaxation time is shorter than the mw period, and EEDF of these electrons responds the time evolution of the electric field immediately (see Sec. II). Other electrons fill only the average electric field defined by (4).

Figure 7(b) shows that for electrons having energy in the vicinity of $\epsilon e$ ≈ 1.5 eV and $\epsilon e$ ≈ 4 eV and $\epsilon e$ > 20 eV, one has $\lambda \epsilon /\Lambda $ < 1. Then, the EEDF of these electrons is defined by the local electric field. For electrons having other energies, one obtains $\lambda \epsilon /\Lambda $ ≫ 1, i.e., their EEDF is not defined by the local value of electric field, i.e., it is of non-local nature.

EEDF at the distance of 0.1 × $dgap$ from the electrode (Fig. 6(a)) consists of low- ($\epsilon e$ < 5 eV) and high-energy ($\epsilon e$ > 5 eV) parts. Both these parts are non-stationary and populated when electric field in that location is the largest. This is obtained during the sheath formation and the reversal of electric field. The boundary $\epsilon e$ ∼ 5 eV is explained by the electron energy losses to vibrational excitation of N_{2} as well as the peak value of electron momentum transfer cross section (Fig. 1(a)). In the latter case, one obtains $\lambda \epsilon /\Lambda $ ∼ 1 (Fig. 7(b)). For electrons with 5 eV < $\epsilon e$ < 20 eV, Fig. 7(b) shows $\lambda \epsilon /\Lambda $ ≫ 1, i.e., they are able to cross the entire interelectrode gap. This explains the population of high-energy tail of EEDF. Here, it is important to note that the region 5 eV < $\epsilon e$ < 20 eV brackets the ionization threshold of N_{2} (15.6 eV). Thus, the non-local electrons are able to ionize N_{2} far from the region of their acceleration, i.e., the sheath (Fig. 4(b)).

One can conclude from Fig. 6(b) that EEDF at 0.2 × $dgap$ from the electrode consists of three parts. The first part consists of electrons with $\epsilon e$ < 1.5 eV. This threshold value corresponds to the peak value of vibrational excitation cross section (Fig. 1(b)). Electrons with $\epsilon e$ < 1.5 eV fill only the average electric field because for them *ω*/*ν* ≫ 1 (Fig. 7(a)). Therefore, this part of EEDF is stationary.

The second group of EEDF seen in Fig. 6(b) has the energy 1.5 eV < $\epsilon e$ < 4 eV. This group is populated during the sheath formation near the closest electrode and depopulated during the sheath reversal. These electrons can only experience elastic collisions (Fig. 1). One can conclude from Fig. 7(a) that these electrons fill only the average electric field because for them $\omega /\nu $ ≫ 1. At the same time, Fig. 7(b) shows for these electrons $\lambda \epsilon /\Lambda $ > 1, i.e., they cannot be confined in this energy range.

The third group of EEDF is the high-energy tail which maintains the plasma in the interelectrode gap. Figure 6(b) shows the asymmetry of this high-energy tail. It is populated by the electrons being accelerated during the sheath formation or the electric field reversal in the vicinity of the electrode. For these electrons, we obtain $\lambda \epsilon /\Lambda $ ≫ 1, i.e., they can reach the location 0.2 × $dgap$ conserving the shape of their EEDF. The asymmetry is explained by the fact that during the sheath formation, electrons move in the small accelerating electric field which is present in the plasma bulk. Thus, they gain additional energy. However, during the sheath reversal, backscattered electrons move against the electric field. Therefore, electrons are decelerated while they move toward the position 0.2 × $dgap$. This explains the asymmetry seen in Fig. 6(b).

The EEDF obtained in the center of the interelectrode gap (Fig. 6(c)) exhibits similar trends with the EEDF obtained at 0.2 × $dgap$ (Fig. 6(b)). Thus, the processes responsible for the shape of EEDF in these two locations are the same. The only difference is the value of the largest energy obtained at 0.2 × $dgap$ and 0.5 × $dgap$. This is explained by the electron energy relaxation during their propagation toward the center.

Thus, we conclude from Fig. 6 that part of the EEDF ($\epsilon e$ < 20 eV) in the atmospheric-pressure N_{2} exhibits non-local behavior in the entire interelectrode gap. The high-energy tail of EEDFs defined by the local electric field is populated by the electrons being accelerated in the vicinity of the electrodes during the sheath formation and the field reversal due to drag force on electrons.

### B. Discharge in argon

Argon is atomic gas, i.e., it does not have vibrational low-energy states. In addition, its momentum transfer cross section differs significantly from that of N_{2} due to Ramsauer effect (see Fig. 1(a)). Figure 1(a) shows that in the energy range 1–5 eV, where the main part of electrons is populated, momentum transfer cross section of N_{2} is ∼1 order of magnitude larger than that of Ar. Taking into account these factors, one can expect significantly different plasma parameters and electron kinetics in Ar and N_{2}.

Figures 8–11 show the results of simulations obtained for the atmospheric-pressure Ar, mw frequency *ν* = 4 GHz, and $Ubr$ = 100 V. The peak plasma density obtained at these conditions is ∼3.5 × 10^{20} m^{−3}, and the average electron energy in the plasma bulk is ∼3.5 eV which are larger than that in N_{2} gas (compare Figs. 8(a) and 2(a)). This is explained by two factors.

On the one hand, the ionization cross section in Ar is larger than in N_{2} (Fig. 1(b)). Therefore, electron having some fixed energy produces more electron–ion pairs in Ar than in N_{2}. On the other hand, the absence of low-energy vibrational excitations and dissociations in Ar increases the average electron energy which also increases the plasma density. In addition, the increase in plasma density results in the decrease in the sheath thickness. The latter decreases the number of electron-neutral collisions in the sheath and increases the electron Joule heating. Indeed, the comparison between Figs. 2(c) and 8(c) shows comparable electron heating in both gases in spite of the much larger sheath voltage for discharge in N_{2} gas.

Figure 8(c) shows that there is no electric field reversal in Ar gas. Moreover, one can see the electron deceleration near the grounded electrode. The latter means that the plasma potential is positive with respect to the grounded electrode, i.e., the sheath is formed in order to confine electrons. This result is in agreement with the conclusions of Ref. 18. Namely, the electron momentum transfer cross section in Ar is a decreasing function of energy for $\epsilon e$ < 10 eV (Fig. 1(a)). Also, this cross section in Ar is much smaller than in N_{2} (Fig. 1(a)). Therefore, the electron drag force due to elastic collisions is also a decreasing function. The comparison between Figs. 4(a) and 11(a) shows that indeed the number of elastic collisions in N_{2} is much larger than in Ar. Also, the spatiotemporal distribution of these collisions is significantly different in both gases. This is explained by the larger electric field obtained in Ar plasma bulk and by the absence of vibrational energy losses.

The distribution of electrons having energy in different range is shown in Fig. 9. The comparison between Figs. 9(a) and 9(b) shows comparable numbers of electrons with 0 eV < $\epsilon e$ < 5 eV and 5 < $\epsilon e$ < 15 eV. This is explained by a high value of the energy threshold of the first electronic level of Ar ($\epsilon ex$ = 11.5 eV), i.e., electrons with $\epsilon e$ < $\epsilon ex$ experience only momentum transfer collisions. Figure 9(c) shows that the number of electrons which are able to ionize the Ar gas is much smaller than the number of electrons from the first two groups (Fig. 9(b)). Also, it is important to note the homogeneous spatial distribution of these electrons. The number of ionizations has the peak value near the plasma sheath edge (Fig. 10(b)).

The comparison between Figs. 3 and 9 shows different spatial distributions of electrons having energy >5 eV. This is explained by the electron energy losses for vibrational excitation and electric field reversal near the grounded electrode in N_{2} microdischarge.

Time evolution of the EEDF in three different locations of the interelectrode gap is shown in Fig. 11. These functions differ significantly from those obtained in N_{2} gas (Fig. 6), which is explained by the additional channels of inelastic energy losses in molecular gas and different shapes of electron-neutral cross sections (Fig. 1).

In the plasma bulk, one can distinguish two groups of electrons forming EEDF. The first group forms the steady-state body of EEDF for $\epsilon e$ < $\epsilon ex\u2009$ (Figs. 11(b) and 11(c)), while the population of the second group follows the mw electric field. This is understood from the ratios $\omega /\nu $ and $\lambda \epsilon /\Lambda $ shown in Fig. 12. One can see that below the excitation threshold, $\omega /\nu $ ≫ 1 and $\lambda \epsilon /\Lambda $ ≫ 1. Thus, the electron energy does not respond to the oscillating electric field, and these electrons fill only the effective electric field (4). However, these electrons are in the non-local regime, because their energy relaxation length is much larger than the distance between electrodes. These electrons are accelerated in this effective electric field and populate the EEDF in the center of the interelectrode gap (Fig. 11(c)), where electric field is the smallest.

Figure 12 shows for electrons with $\epsilon e$ > $\epsilon ex$, $\omega /\nu $ ≪ 1, and $\lambda \epsilon /\Lambda $ ≪ 1. Thus, energetic electrons do respond to the mw electric field, because their energy relaxation time is shorter than the mw period. This explains the modulation of the high-energy tail of EEDF seen in Fig. 11. Also, note that this tail of EEDF is defined by the local electric field.

EEDFs shown in Fig. 11 allow us to conclude that in Ar gas, only the electrons whose energy is below the electronic excitation threshold are in non-local regime. This differs significantly between the electron kinetics of Ar and N_{2} gases. In N_{2} gas, electrons with the energy in the range 5 eV < $\epsilon e$ < 20 eV exhibit non-local behavior (see Sec. III A), i.e., non-local electrons are able to generate plasma. In Ar gas, non-local electrons cannot ionize the gas. However, they are responsible implicitly for the electron heating in the plasma bulk. Namely, the penetration of these electrons into the plasma bulk increases the electron temperature $Te$. Since the ambipolar electric field in the unmagnetized plasma bulk is proportional to the electron temperature,^{19} the increase in $Te$ results in the increase in $E$ in the plasma. This leads to the heating of plasma electrons.

## IV. CONCLUSIONS

Electron kinetics in the microwave (4 GHz) atmospheric-pressure nitrogen and argon microdischarges was studied using a self-consistent one-dimensional Particle-in-Cell Monte Carlo collisions model. Several key differences between these two microdischarges were obtained.

The reversal of electric field near the instantaneous grounded electrode (i.e., formation of an inverted sheath) was obtained in the nitrogen microdischarge. This influenced significantly the heating of electrons and, as a consequence, the generation of plasma. The reversal was caused by the collisional drag force on electrons.

The electron energy distribution function in nitrogen consisted of three main groups. The energy thresholds of these groups were defined by the energy of vibrational and electronic excitation. We obtained that the electrons whose energy is below 20 eV exhibit non-local behavior. These electrons were accelerated in the sheath and in the reversed electric field, and were responsible for the plasma generation. In addition, the energy relaxation time of electrons having energy around 1.5 eV and 4 eV was shorter than the microwave period. Therefore, these electrons responded immediately to the microwave field. Other electrons filled only the average electric field.

The reversal of electric field was not obtained in the argon microdischarge. The electrons were mainly heated due to their acceleration in the sheath. Electron energy distribution function consisted of two groups. The first group consisted of electrons whose energy is below the energy of the first electronic level. The energy relaxation time of these electrons was longer than the microwave period. Therefore, they did not respond to the microwave electric field but filled only the effective field. In addition, these electrons exhibited non-local behavior. The second group had the energy larger than the threshold of electronic level excitation. The energy relaxation time of these electrons was shorter than the microwave period. Therefore, they responded to the microwave electric field. Their energy was defined by the local electric field.

## ACKNOWLEDGMENTS

This work was supported by the Air Force Office of Scientific Research (AFOSR) through a Multi-University Research Initiative (MURI) grant titled “Plasma-Based Reconfigurable Photonic Crystals and Metamaterials” with Dr. Mitat Birkan as the program manager.