The microplasma supported by high microwave frequencies is analyzed using a one-dimensional Particle-in-Cell Monte Carlo collision model. The results indicate that there are two regimes of the discharge operation, which are correlated with different regimes of the electron kinetics of the high-energy tail of the electron energy distribution. We obtain that the regimes of electron kinetics are separated by the frequency $\nu cr$, which is the frequency of collisions between the electrons having the energy equal to the excitation threshold of argon and neutrals. Below $\nu cr$, the tail of the electron energy distribution function is non-stationary and is defined by the instantaneous electric field. This results in very efficient heating of this group of electrons. As a consequence, the increase in the microwave frequency up to $\nu cr$ results in the increase in the plasma density. For driving frequencies larger than $\nu cr$, the tail is stationary, implying less efficient plasma heating. Consequently, a decrease in plasma density is obtained for extremely high driving frequencies.

Microwave driven microplasmas have drawn considerable attention due to their attractive physical features such as high plasma densities, stability at the atmospheric pressure conditions, and chemical reactivity.^{1,2} Arrays of high density microplasmas are of interest for numerous applications such as metamaterials, photonic crystals, and microwave filters (see, for instance, Refs. 1, 3–6 and references therein).

Although there are a few advanced computational models that are capable of simulating coupled microplasma and microwave (mw) phenomena,^{7–9} there is still a need for an understanding of basic physical processes responsible for the microplasma ignition and maintenance, especially at very high driving frequencies. For instance, the fluid and Particle-in-Cell Monte Carlo collision (PIC/MCC) models of mw micro-discharges were compared in Ref. 10. The applicability of the effective field approximation for fluid modeling of high frequency discharges was recently examined in Ref. 11. It was concluded that the classical approaches used in conventional fluid models of atmospheric-pressure discharges must be revised before their application to mw micro-discharges.

In a few recent papers,^{12–14} it was obtained that the small dimensions of mw-driven micro-discharges can be used for better control of the electron energy distribution function (EEDF). This cannot be achieved in traditional barrier discharges used for atmospheric-pressure plasma processing. The EEDF control can be achieved because the electron energy relaxation length exceeds the size of microplasma. As a consequence, the electron being generated and accelerated in one plasma sheath brings high energy to the opposite sheath.

In the present paper, the microplasmas maintained by high microwave driving frequencies (in the range of $\nu =$ 10–300 GHz) are analyzed. Such high-frequency microwaves can be generated using various resonant structures such as ring resonators^{15} or gyrotrons.^{16} In the model, the working gas is argon at a pressure of 10^{5} Pa and a temperature of 300 K. The interelectrode gap is $dgap=$ 100 *μ*m, which is much smaller than the wavelength of mw at the frequency of 300 GHz ($\lambda =$ 1 mm). This allows one to use the one-dimensional PIC/MCC model in the electrostatic approximation in which the voltage $Ut=U0\xb7sin(2\pi \nu t)$ is applied to the left electrode. This model was earlier applied in our studies^{13,17} for the analysis of the microplasma generation by mw having the frequency in the range of $\nu =$ 1–10 GHz.

Gregorio *et al.*^{18} showed that for the frequency of ∼200 GHz, the microplasma can be supported by the voltages as low as ∼2–5 V, i.e., by the voltages much smaller than the excitation energy of Ar. In order to be able to capture such an effect, we updated the model of Ref. 17 by adding the ion conversion reaction Ar^{+} + Ar → Ar_{2}^{+} and the generation of metastable atoms Ar^{*} and their consecutive ionization Ar^{*} + *e* → Ar^{+} + 2*e*. The ions Ar_{2}^{+} were modeled as the macro-particles, while species Ar^{*} were modeled in the hydrodynamic approximation for which the diffusion equation was solved. Also, the electron-ion recombination for both Ar^{+} and Ar_{2}^{+} was considered. The secondary electron emission coefficient due to the ion impact was set equal to 0.2 for both electrodes. In all simulations presented below, we seeded the interelectrode gap with the quasi-neutral homogeneous plasma having the density of 10^{19} m^{−3}. The initial density of Ar^{*} was also set homogeneous and equal to 10^{19} m^{−3}. Such dense plasma can be generated during the breakdown stage when much higher voltages are applied.^{17}

The influence of the mw frequency on the time averaged profiles of the electron density, Ar^{*} density, and electrostatic potential is shown in Fig. 1. Figure 2(a) shows the time-averaged Joule heating profile (i.e., the power density deposited from the electric field to the plasma electrons) and Fig. 2(b) shows the time-averaged electric field. These results were obtained for $U0=$ 25 V. The simulations have shown that the discharge cannot be supported by smaller voltages for $\nu >$ 100 GHz even if the initial plasma density is >10^{20} m^{−3}. In such a case, the largest electron energy obtained in the interelectrode gap only slightly exceeded the ionization energy of Ar^{*}. Therefore, the ionization rate was smaller than the recombination one.

One can see from Fig. 2(a) that the increase in the mw frequency from 10 GHz to 50 GHz results in the increase in the electron density ($ne$), while the increase from 50 GHz to 300 GHz results in the drastic decrease in $ne$. The same trend is obtained for the Ar^{*} density [Fig. 1(b)], the time-averaged potential [Fig. 1(c)], and the power deposited to the electron component of plasma [Fig. 2(a)]. The simulations have shown that the dominant ions in the bulk of quasi-neutral plasma are the Ar_{2}^{+}ions which are generated by the ion conversion reaction. The Ar^{+} ions are dominant only in the vicinity of the electrodes for $\nu \u2264$ 50 GHz.

Figure 1(b) shows the non-homogeneous distribution of Ar^{*} density for $\nu \u2264$ 50 GHz [see Fig. 1(b)] and a flat profile near the center of the gap for $\nu >$ 50 GHz. The comparison between Figs. 1(b) and 2(a) allows one to conclude that the profiles of the Ar^{*} density and the power deposited into electrons demonstrate similar features. The peak of both quantities near the electrodes for 10 and 50 GHz is associated with the secondary electrons emitted from the walls. As it follows from Fig. 1(c), these electrons gain an energy of ∼50 eV in the sheaths. Each of these energetic electrons can produce ∼2–3 excited atoms in the vicinity of the cathode sheath edge. The peak power deposited into electrons seen in the vicinity of the sheaths [Fig. 2(a)] is explained by the highest electric field in the sheaths. For frequencies of 10 and 50 GHz, this electric field is ∼2 orders of magnitude larger than the electric field in the plasma bulk.

The simulation results have shown that the main mechanism of Ar^{+} generation depends on the driving frequency. For 10–50 GHz, the main reaction of Ar^{+} generation in the sheaths is the direct electron impact ionization of Ar. This is obtained because the secondary emitted electrons gain energy in the sheaths exceeding the ionization threshold of Ar (see discussion below). In the plasma bulk, the dominant mechanism is the two-step ionization through the excitation of Ar and consecutive ionization of Ar^{*}. At higher frequencies, only the two-step ionization occurs because the electrons do not have enough energy for the direct electron impact ionization. Note that the two-step ionization is more efficient, since the excitation threshold of Ar is $\epsilon ex$ ∼11.5 eV and, as a consequence, the density of Ar^{*} is higher than the density of Ar^{+} (∼10^{21}–10^{22} m^{−3} and 10^{20}–10^{21} m^{−3}, respectively). Thus, the ionization of Ar^{*} requires only $\epsilon ex,ion$ ∼4.15 eV. For the conditions of the present studies, the number of electrons having energy $\epsilon e>\epsilon ex,ion$ is much larger than the number of electrons with $\epsilon e>\epsilon ion=$ 15.5 eV.

In our previous paper,^{13} we found an increasing plasma density for the increasing mw frequency from 1 GHz to 10 GHz. It was also observed that the microplasma can be supported even without the secondary electron emission from the walls. These results were explained by the decreasing electron wall losses.^{13} Indeed, at very high mw frequencies, the amplitude of the electron oscillations is defined as $x0\u223cqeEme\nu 2$,^{19} where $qe$ and $me$ are the electron charge and mass, respectively, and $E$ is the electric field. Thus, even in an electric field of ∼10^{6} V/m, the amplitude of electron oscillation does not exceed 1 *μ*m. The trapping of more electrons in the interelectrode gap by the oscillating electric field results in an increase in the number of electron-ion pairs produced by trapped electrons. This is seen in Fig. 1(a) for $\nu \u2264$ 50 GHz but not seen for higher frequencies. Note that for very high driving frequencies, the only process which limits the plasma density is the electron-ion recombination.

In order to understand why a further increase in the mw frequency leads to the decrease in the plasma density, one needs to analyze the electron kinetics. The EEDF at the distance of 25 *μ*m from the left electrode as a function of time is shown in Fig. 3. For $\nu \u2264$ 100 GHz, the EEDFs contain the high-energy electrons ($\epsilon e>$ 20 eV) which are the electrons emitted by the wall due to the ion bombardment. For $\nu >$ 100 GHz, these electrons are absent. Also, Fig. 3 shows that for $\nu >$ 100 GHz, the peak energy of the EEDFs' tail is ∼15 eV, i.e., it is smaller than the ionization threshold of the ground state Ar. These results clearly indicate the change between different regimes of the mw microdischarge operation.

The electron kinetics in mw driven discharges can be classified based on the ratio between the mw frequency and the electron-neutral collision frequency:^{20} (1) if $\nu /\nu coll\u226b$ 1, the electric field varies faster than the energy relaxation occurs; then, the electron energy distribution is defined by the effective electric field $Eeff=E\u03032\nu m\nu m2+\omega 2$, where $E\u0303$ is the average electric field; (2) if $\nu /\nu coll\u226a$ 1, the energy relaxes faster than the time evolution of the applied electric field and the electron energy distribution responds to the instantaneous electric field. Recently, it was found^{14} that the parameter $\nu /\nu coll$ together with the ratio between the electron energy relaxation length and the interelectrode gap ($\Lambda \epsilon /l$) defines the non-stationary and the non-local regimes of the electron kinetics in high-frequency driven discharges.

Earlier, it was obtained^{13} that in mw micro-discharges for $\nu <$ 10 GHz, the low-energy part of the EEDF ($\epsilon e<\epsilon ex$) is stationary since $\nu /\nu coll\u226b\u2009$ 1 for these electrons, while the high-energy part of the EEDF $\epsilon e>\epsilon ex$ is non-stationary. It is of interest to examine the regimes of the electron kinetics at extremely high mw frequencies. In order to do so, Fig. 4 shows the electron-neutral collision frequencies and the electron energy relaxation length as functions of electron energy. Note that the elastic scattering cross-section was multiplied by the factor $\delta =2me/mi$ since in each such collision, the electron loses only a fraction $\delta $ of its energy.

Figure 4(a) allows one to conclude that the electrons having the energy below the excitation threshold do not feel the oscillating electric field for all mw frequencies considered in the present study. Thus, the EEDF of this group is stationary. Based on this observation, one can introduce the critical frequency, $\nu cr$, which separates the regimes of stationary and non-stationary EEDF's dynamics. For the atmospheric-pressure Ar, this frequency is $\nu cr\u223c$ 50 GHz. It is estimated for the electron having the energy $\epsilon e=\epsilon ex$. The high-energy tail of the EEDF ($\epsilon e>\epsilon ex$) does feel the field oscillations. However, as it follows from Fig. 4(a), the threshold energy of non-stationary electron kinetics depends on the mw frequency, namely, it increases for increasing $\nu $. For instance, for 10 GHz, this threshold energy is ∼12.5 eV, while for 300 GHz, it is ∼17 eV. That is, for 10 GHz, the electrons having the energy larger than the ionization threshold of Ar feel the instantaneous electric field, while for much higher driving frequencies, a large fraction of these electrons feel the effective electric field which is much smaller ($Eeff\u2248E\u03032$). This results in less efficient heating of the tail of the EEDF at mw frequencies >100 GHz. In addition, it is important to note from Fig. 2(c) that for $\nu \u2265$ 200 GHz the mw frequency exceeds the electron plasma frequency. This means that the electrons do not feel the oscillating electric field but propagate in the time-averaged field as shown in Figs. 1(c) and 2(b).

Figure 4(b) shows that the energy relaxation length of electrons with $\epsilon e<$ 20 eV is longer than the half of the interelectrode gap. This means that the secondary emitted electrons (if any) are able to reach the center of the gap and dissipate the energy there. Moreover, the energy relaxation length of the electrons having the energy $\epsilon e<$ 6 eV and $\epsilon e>$ 16 eV is even longer than the interelectrode gap. Therefore, the electrons with the energy exceeding the ionization threshold of Ar which are generated in the sheaths generate rather homogeneous plasma in the entire gap [see Fig. 1(a) for 50 GHz]. As it follows from our simulations, such electrons are obtained only for $\nu \u2264$ 50 GHz. For higher frequencies, the sheath voltage is smaller than 15 V [Fig. 1(c)]. Therefore, there are no electrons having the energy larger than $\epsilon ion$. Figure 4 shows that the electrons with $\epsilon e<$ 15 eV are in the local regime and feel only the time-averaged electric field.

To summarize, one can describe the maintenance of the mw microdischarge as follows:

- (1)
For $\nu \u2264\u2009\nu cr$, the high-energy tail of the EEDF feels the instantaneous electric field which results in very efficient heating of these electrons. For the atmospheric-pressure argon, $\nu cr=$ 50 GHz. On the one hand, this leads to the increase in the plasma average energy. On the other hand, the increase in the electron energy results in the increase in the sheath potential. The latter results in the increase in the maximum energy of electrons entering the plasma bulk from the sheath. Thus, the increase in the mw frequency from 10 to 50 GHz results in the increase in the plasma density.

- (2)
For $\nu >\u2009\nu cr$, the EEDF tail feels only the effective electric field which is much smaller than the instantaneous electric field. This leads to the less efficient plasma heating. This results in the decrease in the plasma average energy and, as a consequence, in the decrease in the plasma potential. The decrease in the plasma potential decreases the sheath-plasma voltage which results in the decrease in the secondary emitted electrons energy. Thus, the plasma density decreases for driving frequencies larger than 50 GHz.

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.