In this paper, the contribution of ions in RF permittivity and electrical conductivity of atmospheric pressure micro- and sub-microgaps in both pre-breakdown and post-breakdown regimes is investigated. It is shown that ions are ignorable in post-breakdown conditions, while their role becomes significant in pre-breakdown mode especially for gaps on the right side of Paschen's curve. Also, it is demonstrated that the contribution of ions in RF properties increases by decreasing the operating frequency in comparison with ion-neutral collision frequency.

Electromagnetic (EM) properties of micro and nano-discharge spaces are essential in evaluating their interaction with high frequency waves. Even a few volts over a micro/nanogap can ignite gas discharge which has to be carefully considered in an EM performance study. The high density of charged particles results in lossy region that attenuates EM propagating waves. Also, the propagation speed of waves decreases in such an area because of its high dielectric constant.^{1,2} A typical approach of studying EM waves propagating in a discharge area is by defining its effective permittivity and conductivity based on particles number densities as well as operating frequency.^{3–5}

In atmospheric pressure and under electric field excitation, three main mechanisms, namely, field emission, electron-impact ionization, and secondary electron emission are responsible for inducing gas micro/nano-discharge.^{6–8} Field emission^{9} is due to electron tunneling from the surface of the cathode in the presence of strong electric field. On the other hand, heavy ions approaching the cathode surface result in secondary electron emission^{10} as well as ion-enhanced field emission.^{11} Finally, electron-impact ionization^{6,7} is a gas-phase process due to the collision of high energy electrons with neutrals. It has been shown that contributions of these mechanisms in total discharge depend on the pressure-gap *p* × *d* product. Consequently, in atmospheric pressure, field emission has the dominant role in nanoscale discharges. However, secondary electron emission and electron-impact ionization have important contributions in microgaps and field emission can be almost ignored for gaps larger than about 7 *μ*m.^{8} Both DC and RF voltages can induce gas discharge^{12–14} and the breakdown voltage is reached when the generation rate of species becomes higher than their loss rate. At this point, an avalanche increase in the number of charged particles results in formation of a plasma region. Hence, two distinct operation regimes of pre-breakdown and post-breakdown can be considered in discharge problems.

Since ions are much heavier than the electrons, they move slower and respond later to an applied field. Therefore, in many studies, just electrons are considered for modeling a discharge region as a dielectric medium.^{1,15,16} Although this is true for many cases, it is shown in this paper that ions can also have an important contribution specifically in the pre-breakdown regime, where the density of ions is orders of magnitude larger than that of the electrons due to their lower mobility. It is also demonstrated that under atmospheric pressure, the ions contribution is a strong function of frequency for microwave range and beyond while the contribution of electrons becomes strongly frequency dependent in THz range. To prove these concepts, we apply numerical plasma simulations to characterize the behavior of different gap sizes from 0.5 *μ*m up to 10 *μ*m in the presence of external electric fields in both pre-breakdown and post-breakdown regimes.

One-dimensional DC microplasma simulations in atmospheric pressure and room temperature are performed in this study. The applied discharge simulation technique is called particle-in-cell with Monte Carlo collision (PIC/MCC),^{17,18} which is the combination of PIC and MCC methods. In PIC part, each particle is tracked in every time-step by solving equation of motion incorporating self-consistent field calculations. MCC is a method for modeling particles' collisions. Three included electron-neutral collisions are elastic scattering, excitation, and ionization. However, excited neutrals are not tracked due to their very short lifetimes. Two ion-neutral collisions of elastic scattering and charge exchange are considered, as well. Here, nitrogen gas and copper electrodes with work function equal to 4.7 eV are considered. The field enhancement factor which is a strong function of surface roughness is considered to be equal to 70.^{19} Also, according to the nitrogen ionization energy and copper work function, the secondary electron emission coefficient is assumed to be 0.08.^{6} In all presented results, anode is the left electrode.

Three different gap sizes of 0.5 *μ*m, 5 *μ*m, and 10 *μ*m are considered to cover different regions of the modified Paschen's curve in atmospheric pressure.^{20} For each gap size, first the breakdown voltage, *V _{bd}*, is calculated by PIC/MCC simulation with about 2% uncertainty. Then for each gap size, the micro-discharge problem is solved individually to reach the steady state conditions with five applied voltages of 0.8

*V*, 0.9

_{bd}*V*,

_{bd}*V*, 1.1

_{bd}*V*, and 1.25

_{bd}*V*. This range of voltages provides discharge quantities from pre- to post-breakdown conditions which are then compared from the viewpoint of RF properties.

_{bd}Based on PIC/MCC simulations, DC breakdown voltages are equal to 33.1 V, 191 V, and 355 V for the 0.5 *μ*m, 5 *μ*m, and 10 *μ*m gaps, respectively. Since the steady-state conditions even in post-breakdown regimes are required, the simulations were performed in current-driven mode, where the corresponding values of current densities are found in Table I. The steady-state distributions of electrons' and ions' number densities inside the gaps for different applied voltages are shown in Fig. 1. The quasi-neutral plasma regions, with almost equal densities of electrons and ions, are obviously observed for post-breakdown conditions. While the length of this plasma region is larger for wider gaps, electron/ion number densities decrease by increasing the gap size. It is also observed that for the 0.5 *μ*m case, most of the electrons are concentrated near the cathode as a result of field emission dominancy while this is not true for larger gap lengths. To prove the validity of the breakdown points as well as the pre- and post-breakdown regimes, the electric field strength inside the gaps is shown in Fig. 2. The plasma regions with almost zero electric field are observed in the post-breakdown regime.

Gap (μm)
. | 0.8 V
. _{bd} | 0.9 V
. _{bd} | V
. _{bd} | 1.1 V
. _{bd} | 1.25 V
. _{bd} |
---|---|---|---|---|---|

0.5 | $2.05\xd7107$ | $1.05\xd7108$ | $2.7\xd7108$ | $5.1\xd7108$ | $8.85\xd7108$ |

5 | $3.17\xd7103$ | $1.3\xd7104$ | $1.2\xd7105$ | $1.45\xd7106$ | $9.8\xd7106$ |

10 | $1.85\xd7103$ | $8\xd7103$ | $9.5\xd7104$ | $1.1\xd7106$ | $5.9\xd7106$ |

Gap (μm)
. | 0.8 V
. _{bd} | 0.9 V
. _{bd} | V
. _{bd} | 1.1 V
. _{bd} | 1.25 V
. _{bd} |
---|---|---|---|---|---|

0.5 | $2.05\xd7107$ | $1.05\xd7108$ | $2.7\xd7108$ | $5.1\xd7108$ | $8.85\xd7108$ |

5 | $3.17\xd7103$ | $1.3\xd7104$ | $1.2\xd7105$ | $1.45\xd7106$ | $9.8\xd7106$ |

10 | $1.85\xd7103$ | $8\xd7103$ | $9.5\xd7104$ | $1.1\xd7106$ | $5.9\xd7106$ |

Another important point regarding particles' number densities is that in pre-breakdown mode, ions' density is higher than electrons' as a result of the much larger mobility of electrons. For example, for the 0.5 *μ*m gap in 0.8 *V _{bd}*, the maximum density of ions is 4.6× higher than the maximum density of electrons. This ratio grows to 554× for the 10

*μ*m gap case. Now the question becomes: Can the 50 000× heavier ions influence the RF properties of a discharge region since they outgo the electrons at a rate of 5:1 to 500:1 depending on the gap?

To address this question, the RF electrical conductivity of a discharge gap is considered as an evaluation parameter. In order to derive it in presence of both electron and ions, first the Ampere's law in phasor notation is considered

in which, $H\u0303$ and $E\u0303$ represent magnetic and electric fields, respectively. Also, $Jc\u0303$ is the total conduction current due to both electrons and ions

Here, −*e* is the electron's charge and *n _{e}* and

*n*represent electrons' and ions' number densities, respectively. Also, $ve\u0303$ and $vi\u0303$ are drift velocities of electrons and ions, respectively. By putting (2) into (1) and considering the plasma region as a dielectric medium with complex relative permittivity of

_{i}*ϵ*

_{rpc}, we have

On the other hand, since all the applied forces on the particles are due to E-field and collisions, we can write the velocity terms based on second Newton's law

Here, *ν _{e}* is the electron-neutral collision frequency which is estimated by $\nu e=C\xd7p$, where

*p*is the pressure in Torr and

*C*is a gas dependent parameter which is equal to $4.2\xd7109$ (s

^{−1 }Torr

^{−1}) for nitrogen.

^{6}Finally,

*ν*shows the ion-neutral collision frequency which is assumed to be equal to $8\xd7109$ (s

_{i}^{−1}) for nitrogen.

^{21}In addition,

*m*is electron's mass,

*M*represents ion's mass, and $\omega =2\u2009\pi f$ shows the angular frequency of the propagating wave.

By putting (4) into (3) solving the equation for *ϵ*_{rpc} and considering that $\u03f5rpc=\u03f5rp\u2212j\omega \u03f50\sigma p$, we can extract the effective permittivity and conductivity of discharge regions

Here, *σ _{pe}* and

*σ*are the electrical conductivities due to the electrons and ions, respectively.

_{pi}Since $m=9.11\xd710\u221231$ kg and $M=4.64\xd710\u221226$ kg for nitrogen, ions may have a role in conductivity if $ni\u226b\u2009ne$. Fig. 1 shows that this indeed happens before breakdown especially in large gaps. On the other hand, *σ _{p}* is a dispersive parameter which depends on frequency. Considering electrons' contribution term (

*σ*),

_{pe}*ν*is in the order of 10

_{e}^{12 }s

^{−1}in atmospheric pressure. Therefore, for microwave frequencies, $\omega 2\u226a\nu e2$. Hence, electrons' contribution in electrical conductivity is almost independent of frequency. However, for ions,

*ν*is in the order of 10

_{i}^{10 }s

^{−1}in atmospheric pressure which is comparable to

*ω*in the microwave range. Thus, contribution of ions is a strong function of frequency.

Plots of total electrical conductivities for a sample frequency of *f* = 10 GHz for different gap sizes are found in Fig. 3. It is observed that in all pre-breakdown regimes, conductivities are so low that is possible to neglect the effects of discharge region on the propagation of EM waves. However, the conductivities of quasi-neutral plasma regions in post-breakdown regimes are high enough to attenuate propagating EM waves. It is also observed that *σ _{p}* decreases with increasing gap size. As an example, the maximum and minimum simulated values of

*σ*for the 0.5

_{p}*μ*m gap are 797 (S/m) and 0.54 (S/m), while these are equal to 505 (S/m) and $9.4\xd710\u22125$ (S/m) for the 10

*μ*m case.

In order to quantify the contribution of ions in electrical conductivity, a normalize error is defined as

Fig. 4 shows Δ*σ _{p}* for a sample frequency of

*f*= 10 GHz for all gap sizes. It is observed that ions are contributing a lot more significantly before breakdown. Also, the contribution of ions is proportional to the gap size as a result of larger ionization rates in wider gaps. For instance, at 10 GHz, Δ

*σ*exceeds 30% for the 5

_{p}*μ*m case and 40% for the 10

*μ*m gap while it remains below 0.2% for the 0.5

*μ*m gap size. It is also interesting that Δ

*σ*is almost zero in the quasi-neutral plasma regions where the density of electrons and ions are almost equal while electrons move much faster. Fig. 5 shows Δ

_{p}*σ*at

_{p}*f*= 1 GHz. It is observed that errors are higher now since

*σ*is proportional to

_{p}*ω*

^{−2}.

In summary, the contribution of ions in RF electrical conductivity of atmospheric pressure micro and nanoscale gaps was investigated using PIC/MCC simulations. It was shown that ions can have significant role in pre-breakdown conditions. Since electron-impact ionization is significant in larger gaps, the contribution of ions is also increased by gap length. Also, it was observed that ions' contribution decreases with frequency. Therefore, ions become very important before breakdown for large gaps and at lower frequencies. These conclusions are also valid for the permittivity of the discharge region since the dependences on particles' number densities and the collision frequencies are almost similar compared with the conductivity.

This paper is based upon work supported by the National Science Foundation under Grant No. ECCS-1202095.