We study the voltage-current characteristics of gas discharges driven by field emission of electrons at the microscale. Particle-in-cell with Monte Carlo collision calculations are first verified by comparison with breakdown voltage measurements and then used to investigate atmospheric discharges in nitrogen at gaps from 1 to 10 *μ*m. The results indicate the absence of the classical glow discharge regime because field electron emission replaces secondary electron emission as the discharge sustaining mechanism. Additionally, the onset of arcing is significantly delayed due to rarefied effects in electron transport. While field emission reduces the breakdown voltage, the power required to sustain an arc of the same density in microgaps is as much as 30% higher than at macroscale.

## I. INTRODUCTION

According to the classical Paschen law,^{1} the breakdown voltage of macroscopic discharges scales with the product of gas pressure, *p*, and the interelectrode distance, *d*, with an exponential growth as *p* · *d* → 0. However, for microgaps at *d* < 10 *μ*m, a nearly linear drop in the breakdown voltage as *d* → 0 is observed due to field-emission of electrons enhanced by ions in the cathode region. Experimental,^{2–4} theoretical,^{5,6} and numerical^{7–9} studies quantified this deviation, resulting in the modified Paschen curve.^{10,11}

Electrical breakdown in microgaps is a limiting factor for reliability of Microelectromechanical Systems (MEMS).^{12} This motivated an extensive study of the breakdown in microgaps with *d* ≤ 10 *μ*m.^{13,14} However, microdischarges with significant plasma densities (≈10^{19} 1/m^{3}) at atmospheric pressures with sub-millimeter features have found applications as plasma cathodes, plasma reactors for chemical decontamination, and novel photonic devices.^{15–17} Altering the electrical properties, such as the permittivity and conductivity of microgaps through plasmas, makes microdischarges at atmospheric pressures attractive mechanisms for tunable devices. Here, we quantify the variation of electric properties from dark discharge to arcing for atmospheric microdischarges affected by field emission.

A classical V-I curve for a gas discharge indicates three distinct regimes. The end of dark discharge regime marks the breakdown voltage, after which the voltage steeply decreases with a small increase in the current and reaches almost a constant value within the glow regime. A further increase in current leads to arcing. The voltage-current characteristic curves are expected to reduce in amplitude with decrease in the gap size due to reduced breakdown voltage, as predicted by the modified Paschen curve.^{13} Significant alteration of electrical properties of a gas typically requires an ionization fraction^{15} above 10^{−6}. This narrows down the region of interest in the voltage-current characteristics plot^{18} to the end of Townsend dark regime until arc.

The cathode sheath thickness in a normal glow mode corresponds to the (*p* · *d*)_{min} of the Paschen curve for a given pressure. The (*p* · *d*)_{min} corresponding to minimum breakdown voltage for nitrogen gas from Paschen curve is 0.8 Torr cm. At atmospheric pressure, this results in *d _{min}* = 10.5

*μ*m. Field emission from cathode has been found to be significant for gaps below 10

*μ*m.

^{10}Therefore, for gaps below 10

*μ*m, at atmospheric pressure, the glow mode would be absent leading to a direct transition from dark-to-arc mode. Here, we study how the transition from the low-current Townsend dark discharge to the high current arc is affected by the field emission at gaps below 10

*μ*m through numerical modeling.

Sections II–IV describe the problem statement for this work, followed by the numerical modeling approach and results encompassing the discharge structure, voltage-current density characteristics, and electric property variation.

## II. PROBLEM STATEMENT

The main objective of the present work is to characterize the dark-to-arc transition in the microdischarges with field emission through voltage-current density characteristics. The second goal is to understand the effective change in the electrical properties of the microgap by discharge.

Figure 1 shows a schematic of the important processes that are taken into consideration. For an inter-electrode distance on the order of microns, surface irregularities on the cathode amplify the electric field in their vicinity and act as a source of electrons through field emission, as shown in Fig. 1. The length in the transverse direction is much greater than the gap size for the 1D setup as depicted in Fig. 2, which is a generic electrode configuration readily realizable at microscale.

## III. NUMERICAL METHOD

This section describes the numerical model and the setup used to perform the simulations. One-dimensional analysis is performed as the goal is to characterize the dark-to-arc transition in field emission microdischarges in general and not for a specific configuration. The gaps considered in the present analysis are on the order of micrometers at 760 Torr and 0.026 eV in nitrogen gas. Knudsen number for a 10 *μ*m gap considering electron-neutral elastic collisions is approximately 0.05, which is close to the rarefied regime. Here, the time scale of electron-neutral elastic collisions is on the order of picoseconds. Kinetic approach is preferred in this regime, since the continuum approach breaks down at these time and length scales. Therefore, particle-in-cell method with Monte Carlo collisions (PIC/MCC) is used to simulate the discharge process in the rarefied regime. We use xpdp1,^{19} one-dimensional code for PIC/MCC.^{20,21}

The species that are modeled in the simulations are nitrogen ions ($N2+$) and electrons (*e*). The PIC/MCC method effectively solves the Boltzmann equation for multiple charged species considered in the setup as given by the following equations:

where *C*, *f*, *n*, and **v** are the collision integral, velocity distribution function, number density, and relative velocity, respectively. The subscripts “*e*” and “+” represent electrons and ions, respectively, with *q _{e}* being the charge of an electron. The collisions modeled for electrons are (a) elastic collisions with neutrals and (b) ionization collisions with neutrals. The collisions modeled for ions are (a) binary elastic collisions with neutrals, (b) ionization collisions with neutrals, and (c) excitation collisions with neutrals.

In addition to field emission, another source of electrons at the cathode is secondary electron emission and these are included as boundary conditions. The current density used to generate the electrons emitted from the cathode surface is computed from the Fowler Nordheim correlation^{22,23} given below

where *j _{FN}* is the Fowler Nordheim field emission current density,

*E*is the electric field, $\varphi w$ is the work function of the emitting surface (here, 5.1 eV),

*β*is the field enhancement factor, which is based on the roughness of the emitting surface,

*A*and

_{FN}*B*are the Fowler Nordheim constants, $y=3.79\xd710\u22125\beta E/\varphi w,\u2009v(y)=0.95\u2212y2$, and

_{FN}*t*

^{2 }≈ 1.1. A field enhancement factor of 55.0 was chosen for the simulations.

^{24}The generation of electrons from the electrode surface due to secondary electron emission is characterized by the secondary electron emission coefficient (

*γ*), which was chosen to be 0.05 for the simulations.

The algorithm of PIC/MCC simulations^{25} consists of resolving the electric field from the charge density distribution, followed by the solution of equations of motion for each super particle (a collection of real particles) for a given Lorentz force. The collisions are then modeled using Monte Carlo collisions method. This process is repeated at every time step until the simulation reaches steady state for the problem under consideration. The field in the domain is assumed to be electrostatic in nature and hence the magnetic field is neglected. In the Lorenz force, the only active term in the simulations is *q***E**.

The setup used for 1D simulations is shown in Fig. 2. It consists of two electrodes with direct current supply, 200 *μ*m in diameter separated by a distance, *d*. The gas considered is nitrogen at atmospheric pressure and at a temperature of 0.026 eV. The simulations were performed for 4 different gap sizes, i.e., 10 *μ*m, 5 *μ*m, 3 *μ*m, and 1 *μ*m to cover discharges that are both field emission driven and otherwise. The simulations are performed with a minimum of 10 particles of each species in a cell. The cell size (Δ*x*) is constrained by Δ*x* < *λ _{D}*/2, where

*λ*is the Debye length, to resolve the Debye shielding. Thus, a minimum cell size of 0.01

_{D}*μ*m and a maximum cell size of 0.1

*μ*m are used. The time step is then determined from the smallest time scale that is to be resolved in the simulations. In this case, it is the time taken by the fastest moving particle (electron) to cross one cell. Hence, a time step of 2 × 10

^{−15}s is used for 1

*μ*m gap and 2 × 10

^{−14}s is used for other gaps.

For a given voltage, there could be multiple current values based on the voltage-current characteristics of a discharge. Since in the simulations the initial conditions are as close to reality as possible, the voltage driven simulations would tend towards the Townsend discharge rather than resolving the other regions. Also, the voltage driven simulations are unstable after breakdown. Current driven simulation constrains the current density through the gap and is a better tool to resolve the voltage-current density (V-J) characteristics of the discharge beyond the Townsend regime. Hence, current driven simulations were performed to resolve the dark-to-arc transition (V-J characteristics) appropriately.

A comparison of the breakdown voltage (*V _{bd}*) from the simulations and experiments

^{2}for atmospheric nitrogen gas is shown in Table I. The breakdown voltage comparison is done with an experimental Paschen curve, and the breakdown voltage is computed from the corresponding

*p*×

*d*from experiments done for a gap of 3.4 mm. The breakdown voltage for a given gap for PIC/MCC at atmospheric pressure is found from the V-J plot. At a gap of 10

*μ*m under atmospheric conditions, the field emission effect is negligible. For example, at a current density of 3.2 × 10

^{4}A/m

^{2}, the field emission current density for 10

*μ*m is 9 × 10

^{−8}A/m

^{2}, while it is 1.3 A/m

^{2}for 5

*μ*m gap. Hence, the simulation result agrees with the interpolated experimental results with a relative error of 0.5% for 10

*μ*m gap. However, the results for 5

*μ*m deviate from the experimental results by 17.6%. At a gap of 5

*μ*m, the field emission effect can no longer be neglected.

p × d (Torr cm)
. | Gap size (μm)
. | V_{bd}_{,}_{exp} (V) (Ref. 2)
. |
---|---|---|

0.76 | 3400 | 268.3 |

0.38 | 3400 | 295.9 |

p × d (Torr-cm) | Gap size (μm) | V_{bd}_{,}_{PIC}_{∕}_{MCC} (V) |

0.76 | 10.0 | 269.6 |

0.38 | 5.0 | 243.9 |

p × d (Torr cm)
. | Gap size (μm)
. | V_{bd}_{,}_{exp} (V) (Ref. 2)
. |
---|---|---|

0.76 | 3400 | 268.3 |

0.38 | 3400 | 295.9 |

p × d (Torr-cm) | Gap size (μm) | V_{bd}_{,}_{PIC}_{∕}_{MCC} (V) |

0.76 | 10.0 | 269.6 |

0.38 | 5.0 | 243.9 |

## IV. RESULTS

Results from the PIC/MCC simulations detailing the discharge structure, voltage-current density characteristics, and variation of electric properties for multiple microgaps are presented in this section. The simulations are assumed to reach steady state, when the ion and electron number densities in the gap tend to a constant value. Since the simulations are current driven, the voltage across the gap should also tend to a constant value. This voltage is referred to as *V _{steady}*. Figure 3 illustrates the time variation of the voltage and the number of ions across the 10

*μ*m gap at a current density of 3.2 × 10

^{4}A/m

^{2}. The results presented in this section correspond to the simulations that reached steady state, similar to the simulation presented in Fig. 3.

### A. Discharge structure

The discharge structure, i.e., spatial variation of electron number density and nitrogen ion number density for gaps of 10 *μ*m, 5 *μ*m, and 1 *μ*m, is shown in Fig. 4. The discharge is completely positively charged in the gap for 10 *μ*m to 5 *μ*m sizes until a current density of 3.2 × 10^{4} A/m^{2} and the quasi-neutrality is observed in the gap at 3.2 × 10^{5} A/m^{2}, indicating that the transition occurs in this range. This shift is observed to take place at the transition from the dark to arc discharge. However, the plasma is observed in the 1 *μ*m gap at a current density of 3.2 × 10^{7} A/m^{2}. This can be attributed to the mean free path of the electron neutral collisions in atmospheric nitrogen being close to 1 *μ*m. Therefore, more number of electrons are required to generate a plasma in 1 *μ*m gap than in case of the other gaps. Thus, by lowering the pressure, the transition from dark to arc would be even more delayed due to increase in the mean free path.

The extent of quasi-neutrality in the gap expands as current density is increased. The extent of the quasi-neutral plasma in 10 *μ*m gap increases from 2.3 *μ*m for a current density of 3.2 × 10^{5} A/m^{2} to 7.8 *μ*m for a current density of 3.2 × 10^{6} A/m^{2}. The cathode sheath thickness for glow discharge is determined by the (*p* × *d*)_{min} from the Paschen curve to achieve a stable discharge under optimum operating conditions. But, the cathode sheaths seen in Fig. 4 are much thinner than the corresponding *d _{min}* (10.5

*μ*m), indicating the absence of the normal and abnormal glow mode. The current density required to generate a quasi-neutral plasma in a 1

*μ*m gap is almost two orders of magnitude higher than that of 10

*μ*m gap. The relative size of the plasma with respect to the gap size is higher in the 10

*μ*m gap than the other gaps for a given current density. For the parameters considered here, the gaps below 5

*μ*m are found to be field emission driven.

### B. V-J characteristics

The variation of voltage, electron number density, and ion number density with the current density for nitrogen gas is presented in this section. The voltage-current characteristics of discharges help in identifying the breakdown voltage for a given pressure and gap between the electrodes. Although the simulations are current driven, for 1D simulations, a more appropriate parameter for comparison is the current density. Discharges for different currents through multiple gaps were simulated at atmospheric pressure for 10, 5, 3, and 1 *μ*m. The simulations performed for the present work do not consider Coulomb collisions. Hence, the simulations were performed for current densities with a fraction of ionization below 10^{−3}, where the Coulomb collisions are negligible.^{26}

V-J characteristics along with the electron and ion number density variation with the current density, i.e., n-J characteristics for multiple microgaps, are shown in Figure 5. The number densities are spatially averaged over the entire gap. The transition from a dark discharge to an arc is accompanied by breakdown of gas and can be clearly seen in Figs. 5(a)–5(d) by a drop in voltage accompanied by a sudden increase in the electron number density.

The voltage drop in the cathode sheath of a glow discharge is on the order of hundreds of volts, while that of arc discharge is on the order of tens of volts. This is because cathode layer in a glow discharge ensures self-sustenance of plasma by multiple ionization collisions of electrons produced by secondary electron emission. However, in arc discharge, the cathode fall voltage is only on the order of ionization potential as the cathode emission mechanisms are strong enough to sustain high currents. As described by Raizer,^{26} for glow discharges, *j _{e}*

_{,}

_{cathode}/

*j*=

_{T}*γ*/(

*γ*+ 1) and for arcs,

*j*

_{e}_{,}

_{cathode}/

*j*≥ 0.6,

_{T}^{26}where

*j*is the total current density and

_{T}*j*

_{e}_{,}

_{cathode}is the electron current density at cathode. Here, in case of atmospheric discharges in microgaps, field emission tunneling of electrons becomes significant

^{13}and is the cathode emission mechanism responsible for transition to arc.

Since field emission is the important cathode emission mechanism here, we study the variation of *j _{FN}*/

*j*with total current density in Figure 6. If the discharge was transitioning to glow,

_{T}*j*

_{e}_{,}

_{cathode}/

*j*would stay constant at ∼0.05 for glow. However, as can be seen in Fig. 6,

_{T}*j*/

_{FN}*j*> 0.1 above a current density of 3 × 10

_{T}^{5 }A/m

^{2}for 3–10

*μ*m gaps indicating the transition to arc regime. Furthermore,

*j*/

_{FN}*j*> 0.6 above a current density of 3 × 10

_{T}^{6 }A/m

^{2}, which is the arc regime. In case of 3

*μ*m and 1

*μ*m gaps,

*j*/

_{FN}*j*is also high in the dark regime. This is because of increased field emission at smaller microgaps. However, the ionization fraction is 10

_{T}^{−6}, and there is no quasi-neutral plasma. The voltages in arc regime as shown in Fig. 5 range from 10 to 40 V with a plasma number density ∼10

^{22 }m

^{−3}, asserting that this is arc. The falling V-J characteristic due to field emission shown in Fig. 5 is also an indication of the transition to arc.

The comparison of the V-J characteristics in Fig. 5 clearly shows that the breakdown voltage decreases with decreasing gap size. The reason for this is that the field electron emission becomes significant at these microgaps, especially below 5 *μ*m. Hence, a relative drop of 75% in the breakdown voltage from 5 *μ*m to 1 *μ*m is observed, while the relative drop in breakdown voltage from 10 *μ*m to 5 *μ*m is only 10%.

Although the breakdown voltage is significantly lower for 1 *μ*m gap, the transition from dark to arc discharge occurs at a higher current density, more than an order of magnitude (20 times) higher than that of 10 *μ*m gap. This is because, the mean free path of electrons at these conditions is comparable to the gap size leading to very few ionization collisions. The voltage in the arc mode drops only by 14% from 10 *μ*m gap to 1 *μ*m gap. This insensitivity can be attributed to the plasma with approximately the same number densities in both gaps. The number density characteristics follow a similar trend for different gap sizes with respect to each other. As the current tends to higher values, almost the entire gap becomes quasi-neutral.

The variation of the power density required to sustain the discharge with varying current densities at multiple gap sizes is illustrated in Fig. 7. The computed power density values vary from 1 W/mm^{3} to 10 MW/mm^{3}, which correspond to 10 mW to 10 W for an electrode disc with a radius of 100 *μ*m. There is a distinct change in the slope observed at the transition from dark to arc regime. Although the power density required for 1 *μ*m is higher than the 10 *μ*m gap in the dark regime, the power requirement is lower and this can be attributed to the significant field electron emission. The major source of electrons in the dark discharge for gaps below 5 *μ*m is the field emission. The power density for these gaps in the dark regime is observed to be almost independent of the gap size. However, during and after the transition, they deviate from each other.

Townsend discharge has a power requirement below 1 W and is extremely important for applications that require a positive space charge with ionization fraction on the order of 10^{−6}. In the dark regime, the power for 10 *μ*m gap discharge is 74.2% higher than the power required for 3 *μ*m gap discharge. However, in the arc regime, the power for 3 *μ*m gap is 12.2% higher than that for 10 *μ*m gap. Power density requirement is more than an order of magnitude higher for 1 *μ*m gap in comparison to 10 *μ*m gap.

### C. Electrical properties

The electrical property variations in the gas induced by plasma that are considered here are the electrical conductivity and permittivity. The relative permittivity, *ε _{r}*

_{,}

_{p}, and the electrical conductivity,

*σ*, of the medium due to the presence of the plasma in a radio frequency (RF) field are given by the following equations:

_{p}^{27}

where *ω*, *ν _{e}*,

*ν*

_{+}, and

*ε*

_{0}are the radio-frequency oscillations in the medium, the frequency of momentum transfer of the electrons to neutrals, the frequency of momentum transfer of the ions to the neutrals, and the permittivity of free space, respectively.

The variation in the electrical properties considered is directly proportional to the number densities of electrons and ions in the gap. A fraction of ionization 10^{−3} in the arc discharge regime helps in attaining a stable atmospheric discharge. At atmospheric pressure, $\nu e2\u226b\omega 2$. Under these conditions, the dependence of the electric permittivity and conductivity as shown in Eqs. (4) and (5) is not on the RF but on the momentum transfer frequency. The momentum transfer frequency is on the order of a trillion Hertz at atmospheric pressure; however, if the pressure is reduced, it decreases almost linearly.^{26} When the pressure is on the order of few Torr, the RF field variation can affect the electric permittivity variation significantly.

Figure 8 shows the variation of change in relative permittivity and electric conductivity in microgaps under a RF field of 1.0 GHz for atmospheric pressure nitrogen.^{28,29} The relative permittivity of plasma can be negative, and to depict its variation with current density, the change in the relative permittivity (1 − *ε _{r}*) is plotted in Fig. 8(a). As the current density increases, the change in the electric properties also increases, because of the increase in the number density of ions and electrons. The maximum change as observed in Figs. 8(a) and 8(b) is for the 10

*μ*m gap. At a current density of 3.2 × 10

^{4}A/m

^{2}, the change in relative permittivity is 0.2% for 10

*μ*m gap and 0.05% for 1

*μ*m gap. At a current density of 3.2 × 10

^{5}A/m

^{2}, the change in relative permittivity is 7.5% for 10

*μ*m gap and 0.2% for 1

*μ*m gap. The highest increase in electric conductivity of the gap due to plasma at a current density of 3.2 × 10

^{4}A/m

^{2}is 2.3 × 10

^{−4}S/m for 10

*μ*m gap. Hence, for tunability at atmospheric pressure, arc is the most effective regime.

## V. CONCLUSIONS

One-dimensional PIC/MCC simulations for microdischarges from 1 to 10 *μ*m gaps in nitrogen at 760 Torr and 0.026 eV at varying current densities have been performed. V-J characteristics along with charge particle density variation with current density have been presented. Along the current density, the transition to arc is delayed in 1 *μ*m gap by 20 times in comparison to 10 *μ*m gap. Field emission promotes the discharge by lowering the breakdown voltage. The delay in the dark-to-arc transition can be attributed to the rarefied effect in the electron transport when the gap size reaches the mean free path for electrons. The transition would be further delayed for lower pressures. The power required to sustain a discharge in arc regime at 1 *μ*m is 30% higher than that for 10 *μ*m with approximately the same discharge number density. A significant change in the electric permittivity and conductivity was only found beyond the Townsend discharge regime. Therefore, to achieve a considerable change (0.01) in the electrical properties, arc discharge is the most suitable region especially for current densities above 1.6 × 10^{5} A/m^{2} and microgaps above 5 *μ*m.

## ACKNOWLEDGMENTS

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