## 4 Simulation
### Governing Equations
#### 4.3.1 The Electromagnetic Field Model  
The electric field in the microwave model satisfied the following equation deduced from the [PERSON]'s equations. The wave equation for this case assumed that the time variation of the electric and magnetic field components was harmonic and the medium was non-magnetic [43]. \[\
abla\times[\mu_{r}^{-1}(\
abla\times E)]-\left(\omega^{2}\varepsilon_{0} \varepsilon_{r}-j\omega\sigma\right)\,E=0 \tag{1}\]  
This Helmholtz equation for the electric field was solved simultaneously in the waveguiding structure and in the plasma chamber. Here, \(E\) denotes the electric field(HF field), \(k_{0}\) is the wave number in free space (\(k_{0}=\omega\sqrt{\mu_{0}\varepsilon_{0}}\)), \(\omega\) is the angular frequency (\(2\pi\times\) 2.45 GHz), \(\varepsilon_{0}\) and \(\mu_{0}\) are the free space permittivity (\(8.854\times 10^{-12}\)\(\left[\mathrm{F}/\mathrm{m}\right]\)) and free space permeability (\(4\pi\times 10^{-7}\)\(\left[\mathrm{N}/\mathit{A}^{2}\right]\)), \(j\) is imaginary unit, \(\sigma\) is conductivity, respectively. \(\varepsilon_{r}\) is the relative permittivity depending on the property of the materials. As the material property, the relative permittivity of the quartz window is \(\varepsilon_{r}^{d}=4\). The alumina used in TE\({}_{11}\) mode had a relative permittivity of \(\varepsilon_{r}^{d}=9\), the relative permittivity of copper is \(\varepsilon_{r}^{e0}=1\), the relative permittivity of air is \(\varepsilon_{r}^{air}=1\), and the relative permittivity of plasma is expressed as \(\varepsilon_{r}^{p}\) in the following equations. \[\varepsilon_{r}^{p}=1-\frac{\omega_{pe}^{2}}{\omega(\omega-iv_{en})}, \tag{2}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Reaction Formula** & **Type** & \(\Delta\varepsilon\) (\(eV\)) & \(\sigma_{e}\kappa\)\(k^{\prime}\) \\ \hline e + Ar \(\rightarrow\) e + Ar & Elastic scattering & - & [38] \\ \hline e + Ar \(\rightarrow\) e + Ar\({}^{\star}\) & Excitation & 11.55 & [38] \\ \hline e + Ar\({}^{\star}\)\(\rightarrow\) e + e + Ar\({}^{\star}\) & Step-ionization & 4.16 & [39] \\ \hline e + Ar \(\rightarrow\) e + e + Ar\({}^{\star}\) & Direct-ionization & 15.76 & [38] \\ \hline Ar + Ar\({}^{\star}\)\(\rightarrow\) Ar + Ar\({}^{\star}\) & Scattering & - & [40] \\ \hline Ar + Ar\({}^{\star}\)\(\rightarrow\) Ar + Ar\({}^{\star}\) & Charge exchange & - & [40] \\ \hline e + Ar\({}^{\star}\)\(\rightarrow\) e + Ar & Quenching & - & [41] \\ \hline Ar + Ar\({}^{\star}\)\(\rightarrow\) Ar + Ar & Quenching & - & [41] \\ \hline Ar\({}^{\star}\) + Ar\({}^{\star}\)\(\rightarrow\) e + Ar\({}^{\star}\) + Ar & Ionization & - & [41] \\ \hline \end{tabular}
\end{table} 
Table 1: Selected reaction for Argon discharges (\(\Delta\varepsilon\): threshold energy, \(\sigma_{c}\): cross-section, \(k^{\prime}\) : rate coefficient (m\({}^{3}\)/s)) [38; 39; 40; 41]. \[\omega_{pe}=\sqrt{\frac{n_{e}q^{2}}{m_{e}\epsilon_{0}}}, \tag{3}\]  
where \(q\) is the elementary charge and \(m_{e}\) is the mass of an electron. \(\omega_{pe}\) denotes the plasma frequency that depends on the electron density \(n_{e}\), and the \(v_{en}\) is the electron-neutral collision frequency for momentum transfer [44]. \[v_{en}=k_{Ar}^{tot}(\overline{z})\cdot n_{Ar} \tag{4}\]  
\(v_{en}\) is expressed as a function of mean electron energy \(\overline{z}\) and the number of neutral particles densities [34,45]. The expression \(k_{i}^{tot}\) is the total collision rate of neutral species i. By calculating the mean electron energy (\(\overline{z}\)) with a Maxwell distribution function and substituting it into the first moment Boltzmann equation, \(\sigma_{p}\) (plasma conductivity) can be obtained as a function of the plasma parameter. \[\sigma_{p}=\frac{n_{e}q^{2}}{m_{e}(v_{en}+j\omega)} \tag{5}\]  
The unmagnetized plasma conductivity \(\sigma_{p}\) is expressed as a function of \(n_{e}\), \(v_{en}\), and \(\omega\). The conductivity of copper is \(6\times 10^{7}\) S/m.