Analytic theory for field emission driven microscale gas breakdown for a pin-to-plate geometry

Loveless, Amanda M.; Breen, Lorin I.; Garner, Allen L.

doi:10.1063/5.0037724

Decreasing electronics size necessitates better characterization of electron emission at the micro- and nanoscales for applications including microplasmas, micro- and nanoelectromechanical systems, and directed energy. While Paschen's law (PL) has historically predicted breakdown voltage based on the Townsend avalanche, field emission must be incorporated for gap sizes below ∼15 μm. Extensive studies have modified PL to explicitly include field emission for planar geometries; however, many practical experiments use pin-to-plate geometries. We modify a previous theory coupling PL and field emission to account for pin-to-plate geometries by replacing the field enhancement factor, which has been used primarily as a fitting parameter, with the appropriate vacuum electric field. This requires explicitly accounting for the spatial dependence of ionization and non-uniform space charge in Poisson's equation. We derive a breakdown equation of the form previously obtained for planar geometry [Venkattraman and Alexeenko, Phys. Plasmas 19, 123515 (2012)] that agrees well with experimental data with the work function as the fitting parameter. The work function was consistently lower (∼2 eV) than anticipated (∼4.5 eV) but was generally fairly consistent (∼ ± 7%). We then derived closed form solutions in the limit of low ionization, corresponding to the field emission regime, and recovered an analytic solution for a parallel plate geometry in the limit of small gap distance that differed from prior analytic results because of the explicit consideration of spatial dependence in charge density. This theory may ultimately be applied to other nonplanar geometries by applying the appropriate equation for the vacuum electric field.

I. INTRODUCTION

The continuous reduction in the device size of electronics toward micro- and nanoscales makes accurately predicting breakdown voltage V_b for safe and optimum device operation increasingly important. Specifically, predicting V_b is paramount for device reliability for applications in high voltage devices,¹ pulsed power,² switches and switchgear,^3,4 high power microwaves,^5,6 directed energy,⁷ fusion,^8–11 and accelerator applications.¹² Conversely, combustion^13–16 and various biomedical applications^17–21 motivate accurate predictions of V_b to ensure the generation of the resultant plasmas.

Historically, Paschen's law (PL)^22,23 has been used to predict breakdown voltage, driven by electron avalanches occurring based on electron collisions. However, as the gap distance decreases, PL begins to deviate from experimental values because the strong electric fields generated at smaller gaps reduce the potential barrier at the cathode sufficiently for electrons to tunnel into the gap by field emission.^24,25 These field-emitted electrons ionize the neutral gas near the cathode, creating positive ions. The positive ions subsequently collide with the cathode to create an additional component of secondary emission in the Townsend avalanche and a space-charge electric field that enhances the surface electric field and, thus, the field emission current.^23,25–29 These phenomena combine to increase the total current density in the gap, leading to a higher breakdown electric field for a given voltage and causing V_b to decrease with decreasing gap distance d rather than increasing as predicted by PL.^25,27 Recent analytic theories have derived from first principles that $V_{b} \propto d$ in the field emission dominated regime.²³

While gas, gap distance, and pressure are common parameters of interest for breakdown models, an additional consideration is device geometry. Many original models are derived based on a parallel plate geometry;^23,27,30 however, practical devices often use other electrode shapes, such as spherical^31,32 or pin-to-plate.^33,34 Despite the practical importance of these nonplanar geometries, few theoretical studies have investigated these designs.^30,35

A common issue with previous breakdown models^{23,26,36–38} is how to determine the field enhancement factor a priori to allow for a truly predictive model. The difficulty in achieving this has led to the common treatment of the field enhancement factor as a fitting parameter.²³ One recent study eliminates the need to use the field enhancement factor as a fitting parameter and incorporates a non-uniform electric field.³⁵ Here, we combine the recent work by Fu et al. on electric field non-uniformity³⁵ with prior work deriving microscale gas breakdown for a planar diode coupling field emission and Townsend avalanche^26,36 to obtain a dimensionless breakdown model for microscale gas breakdown for a pin-to-plate geometry, which is more common for typical field emission and breakdown applications due to the greater field enhancement than in planar diodes.^33,34 In the process, we now use the work function as the fitting factor and observe a relatively slight variation (∼15%) across various gap conditions. Also, electrode surface waviness impacts the measured (or effective) work function,³⁹ which may account for some of this variation, particularly for multiple breakdown events that can alter the electrode surface, and even cause significant cratering.³⁴

Section II derives the breakdown model in detail. We compare the theory to experimental data and apply the theory to explore trends as a function of gap distance, pressure, and work function in Sec. III. Section IV provides concluding remarks.

II. MODEL

Figure 1 shows the pin-to-plate geometry considered in this analysis with the pin anode with a tip radius of 0.5 μm at x = 0 and a copper plate as the cathode at x = d set at a gap distance of 1, 5, or 10 μm ± 0.5 μm from the tip.³⁴

FIG. 1.

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Pin-to-plate geometry considered in the analysis, where the pin represents the anode set at x = 0 with a tip radius of r and the plate is the cathode at a gap distance of d from the tip. Processes for electron emission include secondary electron emission, ionization, and field emission, which create the space-charge region near the cathode.

Following an analogous procedure to Ref. 26, we begin by writing Poisson's equation assuming all spatial variation in the electric field E is in the x direction as

\nabla \cdot E = \frac{d E_{t o t} (x)}{d x} = \frac{ρ}{ε_{0}} = \frac{j_{i o n}}{v_{d} ε_{0}},

(1)

where $E_{t o t} (x)$ is the total electric field [ $E_{t o t} (x) = E_{v} (x) + E^{+} (x)$ ⁠, where $E_{v}$ is the vacuum electric field and $E^{+}$ is the space-charge electric field], $j_{i o n}$ is the ion current density, given by

j_{i o n} = \frac{j_{t o t} [\exp (\int_{0}^{d} α (x) d x) - 1]}{\exp (\int_{0}^{d} α (x) d x)},

(2)

where $j_{t o t}$ is the total current density, and

v_{d} = \sqrt{\frac{2 e k T_{g} E_{t o t} (x)}{π m p σ_{c e}}}

(3)

is the ion drift velocity with all terms and variables are described in Table I. Note that the total electric field $E_{t o t} (x)$ is a function of position x in the geometry in Fig. 1, which makes the ionization coefficient $α (x)$ spatially dependent and given by

α (x) = A_{p} p \exp (- \frac{B_{p} p}{E_{t o t} (x)}) .

(4)

TABLE I.

Parameter definitions, values, and units used in the derivation.

Parameter	Definition	Value or equation	Unit
A_p	Material constant of air	15	cm⁻¹ Torr⁻¹
B_p	Material constant of air	365	V cm⁻¹ Torr⁻¹
A_FN	Fowler–Nordheim constant	6.2 × 10⁻⁶	A eV V⁻²
B_FN	Fowler–Nordheim constant	6.85 × 10⁷	$V c m^{- 1} e V^{- 3 / 2}$
$D_{F N}^{'}$	Modified Fowler–Nordheim parameter	$0.95 B_{F N} ϕ^{3 / 2}$	V cm⁻¹
ɛ₀	Permittivity of free space	8.854 × 10⁻¹²	F m⁻¹
e	Electron charge	1.602 × 10⁻¹⁹	C
k	Boltzmann's constant	1.38 × 10⁻²³	J K⁻¹
T_g	Gas temperature	300	K
m	Mass of gas molecule	4.82 × 10⁻²⁶	kg
σ_ce	Charge exchange cross section	10⁻¹⁸	m²
p	Gas pressure	Variable	Torr
x	Position along gap	Variable	m
r	Anode tip radius	Variable	m
ϕ	Effective cathode surface work function	Variable	eV
d	Gap distance	Variable	m
γ_SE	Secondary emission coefficient of cathode from ion impact	Variable	Dimensionless
t²(y)	Approximate correction factor for elliptical integrals in Fowler–Nordheim equation	1.1	Dimensionless
ρ	Ion charge density	$j_{i o n} / v_{d}$	C m⁻³
v_d	Ion drift velocity	cf. (3)	m s⁻¹

Parameter	Definition	Value or equation	Unit
A_p	Material constant of air	15	cm⁻¹ Torr⁻¹
B_p	Material constant of air	365	V cm⁻¹ Torr⁻¹
A_FN	Fowler–Nordheim constant	6.2 × 10⁻⁶	A eV V⁻²
B_FN	Fowler–Nordheim constant	6.85 × 10⁷	$V c m^{- 1} e V^{- 3 / 2}$
$D_{F N}^{'}$	Modified Fowler–Nordheim parameter	$0.95 B_{F N} ϕ^{3 / 2}$	V cm⁻¹
ɛ₀	Permittivity of free space	8.854 × 10⁻¹²	F m⁻¹
e	Electron charge	1.602 × 10⁻¹⁹	C
k	Boltzmann's constant	1.38 × 10⁻²³	J K⁻¹
T_g	Gas temperature	300	K
m	Mass of gas molecule	4.82 × 10⁻²⁶	kg
σ_ce	Charge exchange cross section	10⁻¹⁸	m²
p	Gas pressure	Variable	Torr
x	Position along gap	Variable	m
r	Anode tip radius	Variable	m
ϕ	Effective cathode surface work function	Variable	eV
d	Gap distance	Variable	m
γ_SE	Secondary emission coefficient of cathode from ion impact	Variable	Dimensionless
t²(y)	Approximate correction factor for elliptical integrals in Fowler–Nordheim equation	1.1	Dimensionless
ρ	Ion charge density	$j_{i o n} / v_{d}$	C m⁻³
v_d	Ion drift velocity	cf. (3)	m s⁻¹

Additionally, the Fowler–Nordheim current density^24,26,40 is given by

j_{F N} = \frac{A_{F N}}{ϕ t^{2} (y)} \exp (\frac{{(3.79 \times 10^{- 4})}^{2} B_{F N}}{ϕ^{1 / 2}}) {[E_{t o t} (d)]}^{2} \exp (- \frac{0.95 B_{F N} ϕ^{3 / 2}}{E_{t o t} (d)}),

(5)

where $E_{t o t} (d)$ is the total electric field at the cathode and the total current density is given by

j_{t o t} = \frac{j_{F N} \exp (\int_{0}^{d} α (x) d x)}{1 - γ_{S E} [\exp (\int_{0}^{d} α (x) d x) - 1]} .

(6)

Applying (2) and (3) to (1) yields

\frac{d E_{t o t} (x)}{d x} = \frac{j_{t o t} [1 - \exp (- \int_{0}^{d} α (x) d x)] {(π m p σ_{c e})}^{1 / 2}}{ε_{0} {(2 e k T_{g} E_{t o t} (x))}^{1 / 2}} .

(7)

Additionally, the vacuum (or space-charge independent) electric field $E_{v} (x)$ for the geometry in Fig. 1 is given by³⁵

E_{v} (x) = \frac{2 V}{(2 x + r) \ln (2 d / r + 1)} .

(8)

The positive space charge in the gap incrementally increases the electric field by E⁺.^25,26 Thus, j_FN from (5) becomes the modified Fowler–Nordheim current density, given by

\begin{aligned} j_{F N}^{'} & = \frac{A_{F N}}{ϕ t^{2} (y)} \exp (\frac{{(3.79 \times 10^{- 4})}^{2} B_{F N}}{ϕ^{1 / 2}}) {(E_{v} (d) + E^{+} (d))}^{2} \\ \times \exp (- \frac{0.95 B_{F N} ϕ^{3 / 2}}{E_{v} (d) + E^{+} (d)}) . \end{aligned}

(9)

Assuming $E^{+} (d) ≪ E_{v} (d)$ allows us to take a series expansion of (9) and substitute it into (6) to obtain²⁶

j_{t o t}^{'} = \frac{j_{F N}^{'} \exp (\int_{0}^{d} α (x) d x)}{1 - γ_{S E} [\exp (\int_{0}^{d} α (x) d x) - 1]} = j_{F N} \exp (\frac{D_{F N} E^{+} (d)}{E_{v} {(d)}^{2}}) (1 + \frac{2 E^{+} (d)}{E_{v} (d)}) \frac{\exp (\int_{0}^{d} α (x) d x)}{1 - γ_{S E} [\exp (\int_{0}^{d} α (x) d x) - 1]} .

(10)

At breakdown, the electric field in j_FN is defined as the surface field, or the electric field at the cathode x = d. Strictly speaking, E_tot(x) = E_v(x) + E⁺(x); however, we can make a few observations about the contributions of these terms to streamline our derivation. First, since E⁺ arises due to the ions formed at the cathode, the maximum contribution due to space charge will occur for the surface electric field at the cathode in field emission and the secondary emission coefficient γ_SE in the Townsend avalanche condition.^23,26 The peak E⁺ at x = d is consistent with prior theories for planar microscale gas breakdown, which assumed that the charge density was constant from the cathode to the center of the gap and absent from the center to the anode.^23,26 Here, we relax the requirement of constant charge density from the cathode to the center of the gap, but retain the assumptions of peak space-charge contribution at the cathode and negligible space-charge contribution due to the ions at the anode.

For α(x), including E⁺(x) explicitly in (4) would greatly complicate the solution, particularly since E⁺(x) is at its highest near the cathode and approaches zero at the anode and, more generally, $E^{+} (x) ≪ E_{v} (x)$ throughout the gap. Because the physical importance of E⁺(x) is its influence on field emission at the cathode surface at x = d,^23,26 its impact on E_tot(x) and, consequently, α(x), throughout the remainder of the gap is minimal, we assume E_tot(x) ≈ E_v(x) across the gap for determining α(x). As before,^23,26 the surface electric field at the cathode (x = d) for determining the contribution of field emission must include this space-charge electric field, so E_tot(d) = E_v(d) + E⁺(d). Because of the difficulty in extracting space-charge effects from $E_{t o t} (d)$ ⁠, we cannot readily determine the applied breakdown voltage at the anode $(x = 0)$ $V_{b}$ from the cathode condition. Instead, we again appeal to $E^{+} (x) ≪ E_{v} (x)$ and, more specifically, $E_{t o t} (0) \approx E_{v} (0)$ to estimate V_b at the anode from the vacuum breakdown electric field at the anode, given by

E_{b} (0) = E_{b v} (0) + E^{+} (0) \approx E_{b v} (0) \approx \frac{2 V_{b}}{r \ln (2 d / r + 1)} .

(11)

While $E^{+} (x) ≪ E_{v} (x)$ throughout the gap, we emphasize that E⁺(x) is at its maximum at the cathode and at its minimum [we explicitly assumed $E^{+} (0) \to 0$ ] at the anode, making (11) a more accurate approach for determining $V_{b} .$ Rearranging (11) gives the voltage applied at the anode (x = 0) to achieve breakdown as

V_{b} = \frac{r E_{b} \ln (2 d / r + 1)}{2} .

(12)

Substituting (12) into (8) yields the vacuum electric field across the gap at breakdown as

E_{b v} (x) = \frac{r E_{b v} (d)}{(2 x + r)} = \frac{r E_{b}}{2 x + r} .

(13)

Integrating $- α (x)$ along the gap to substitute into (7) gives

\begin{aligned} - \int_{0}^{d} α (x) d x & = - \int_{0}^{d} A_{p} p \exp [- \frac{B_{p} p}{E_{b} r} (2 x + r)] d x \\ = \frac{A_{p} E_{b} r}{B_{p}} [\exp (- \frac{B_{p} p (2 d + r)}{E_{b} r}) - \exp (- \frac{B_{p} p}{E_{b}})] . \end{aligned}

(14)

From here, we nondimensionalize by³⁶

\begin{aligned} V & = \bar{V} V_{*} = \bar{V} E_{*} d_{*} : d = \bar{d} d_{*}; E = \bar{E} E_{*}; r = \bar{r} d_{*}; x = \bar{x} d_{*}; \\ T & = \bar{T} T_{*}; α = \bar{α} d_{*}^{- 1}, \end{aligned}

(15)

where

\begin{aligned} E_{*} = 0.95 B_{F N} ϕ^{3 / 2}; j_{*} = \frac{A_{F N}}{ϕ t^{2} (y)} \exp (\frac{{(3.79 \times 10^{- 4})}^{2} B_{F N}}{ϕ^{1 / 2}}) E_{*}^{2}; p_{*} = \frac{E_{*}}{B_{p}}, \\ d_{*} = \frac{1}{A_{p} p_{*}}; T_{*} = \frac{π m σ_{c e} B_{p}}{2 e k ε_{0}^{2}} {[\frac{A_{F N}}{ϕ t^{2} (y) A_{p}} \exp (\frac{{(3.79 \times 10^{- 4})}^{2} B_{F N}}{ϕ^{1 / 2}})]}^{2} . \end{aligned}

(16)

Considering the cathode at x = d, we next incorporate space charge into the total current density through the modified Fowler–Nordheim current from (9) as

{\bar{j}}_{t o t}^{'} = \frac{{\bar{j}}_{F N}^{'} \exp (\int_{0}^{\bar{d}} \bar{α} (\bar{x}) d \bar{x})}{{1 - γ_{S E} [\exp (\int_{0}^{\bar{d}} \bar{α} (\bar{x}) d \bar{x}) - 1]}} = \frac{{\bar{j}}_{F N} (1 + 2 {\bar{E}}^{+} (d) / {\bar{E}}_{t o t} (d)) \exp {({\bar{E}}^{+} (d) / {\bar{E}}_{t o t} (d))}^{2} \exp (\int_{0}^{\bar{d}} \bar{α} (\bar{x}) d \bar{x})}{{1 - γ_{S E} [\exp (\int_{0}^{\bar{d}} \bar{α} (\bar{x}) d \bar{x}) - 1]}},

(17)

where ${\bar{E}}^{+} (d)$ represents the enhancement in the electric field at the cathode due to the positive space charge induced by the additional ionization due to the electrons that enter the gap due to field emission. Inserting (17) into the nondimensionalized form of (6) and assuming $\bar{α} \bar{d} ≪ 1,$ which generally holds until transitioning to PL,²³ yields

\frac{d {\bar{E}}_{t o t}}{d \bar{x}} \approx \sqrt{\frac{\bar{p}}{\bar{T}}} \frac{\int_{0}^{\bar{d}} \bar{α} (\bar{x}) d \bar{x} {\bar{E}}_{b}^{2} \exp (- 1 / {\bar{E}}_{b}) (1 + 2 {\bar{E}}^{+} (d) / {\bar{E}}_{b}) \exp ({\bar{E}}^{+} / {\bar{E}}_{b}^{2})}{\sqrt{{\bar{E}}_{v} (\bar{x})} (1 - γ_{S E} \int_{0}^{\bar{d}} \bar{α} (\bar{x}) d \bar{x})},

(18)

where ${\bar{E}}_{v} (\bar{x}) = {\bar{E}}_{b} (\bar{d} + \bar{r}) / (2 \bar{x} + \bar{r})$ ⁠, and we have used ${\bar{E}}^{+} (d) ≪ {\bar{E}}_{t o t} (d)$ to obtain ${\bar{E}}_{t o t} (d) \approx {\bar{E}}_{b}$ for ultimately determining V_b. Solving (18) is challenging due to the spatial dependence of ${\bar{E}}_{v} (\bar{x})$ ⁠, which comes from $v_{d}$ in (3); however, the variation of ${\bar{E}}_{v}^{1 / 2} (\bar{x})$ leads to at most a variation in $v_{d}$ of ∼20% at 1 μm and <10% at ∼0.1 μm. Thus, for the purposes of solving (18), we may neglect this spatial variation and set ${\bar{E}}_{v} (\bar{x}) \approx {\bar{E}}_{b}$ for simplicity. This is reasonable since $γ_{S E} ≪ 1$ (generally considered between 10⁻³ and 10⁻¹)^23,38,41,42 and numerical solutions show that $\bar{α} \bar{d} ≲ 15$ prior to transitioning to PL for gaps ∼10 μm.²³ While this assumption may hinder predictive capability in the transition regime between the field emission driven regime and PL, $\bar{α} \bar{d} ≲ O (1)$ at smaller gaps where field emission clearly drives breakdown,²³ making this assumption more accurate in the more field emission driven regime. Furthermore, sensitivity analysis using error propagation showed that $γ_{S E}$ had minimal impact on V_b when field emission dominated in a parallel plate geometry,⁴³ providing additional justification for neglecting this term for calculations in the field emission regime. Thus, we may neglect the second term in the parentheses of the denominator of (18) since $γ_{S E} \bar{α} \bar{d} ≪ 1$ in the FE dominant breakdown regime, allowing us to rewrite (18) as

\begin{aligned} \frac{d {\bar{E}}_{t o t}}{d \bar{x}} & = - \sqrt{\frac{\bar{p}}{{\bar{E}}_{b} \bar{T}}} {\bar{E}}_{b}^{3} \bar{r} [\exp (- \frac{\bar{p} (2 \bar{d} + \bar{r})}{{\bar{E}}_{b} \bar{r}}) - \exp (- \frac{\bar{p}}{{\bar{E}}_{b}})] \\ \times \exp (\frac{- 1}{{\bar{E}}_{b}}) (1 + \frac{2 \bar{E} +}{{\bar{E}}_{b}}) \exp (\frac{{\bar{E}}^{+}}{{\bar{E}}_{b}^{2}}) . \end{aligned}

(19)

Integrating (19) across the gap (0 to $\bar{d}$ ⁠) yields

\begin{aligned} {\bar{E}}_{t o t} (\bar{d}) = \exp [\frac{- 2 \bar{d} \bar{p} - \bar{r} - 2 \bar{p} \bar{r}}{{\bar{E}}_{b} \bar{r}}] {\bar{E}}_{b} {\exp [\frac{2 \bar{d} \bar{p} + \bar{r} + 2 \bar{p} \bar{r}}{{\bar{E}}_{b} \bar{r}}] \\ - \bar{d} {\bar{E}}_{b} \bar{r} \sqrt{\frac{\bar{p}}{{\bar{E}}_{b} \bar{T}}} ({\bar{E}}_{b} + 2 {\bar{E}}^{+} (d)) \exp [\frac{{\bar{E}}^{+} (d)}{{\bar{E}}_{b}^{2}}] \exp [\frac{\bar{p}}{{\bar{E}}_{b}}] \\ + \bar{d} {\bar{E}}_{b} \bar{r} \sqrt{\frac{\bar{p}}{{\bar{E}}_{b} \bar{T}}} ({\bar{E}}_{b} + 2 {\bar{E}}^{+} (d)) \exp [\frac{{\bar{E}}^{+} (d)}{{\bar{E}}_{b}^{2}}] \exp [\frac{(2 \bar{d} + \bar{r}) \bar{p}}{\bar{r} {\bar{E}}_{b}}]} . \end{aligned}

(20)

This differs from the planar case, which assumed that the space charge was constant from the center of the gap to the cathode (d/2 ≤ x ≤ d), making $E^{+} \propto d$ ⁠.²⁶ Subtracting ${\bar{E}}_{b}$ from both sides of (20) to obtain ${\bar{E}}^{+} (d)$ on the left-hand-side (LHS) and setting $y = {\bar{E}}^{+} (d) / {\bar{E}}_{b}^{2}$ gives

\begin{aligned} 1 & = \frac{(1 + 2 y {\bar{E}}_{b}) \exp [y]}{y} {\exp [\frac{- \bar{r} - \bar{p} \bar{r}}{{\bar{E}}_{b} \bar{r}}] \bar{d} \bar{r} \sqrt{\frac{\bar{p}}{{\bar{E}}_{b} \bar{T}}} {\bar{E}}_{b} \\ - \exp [\frac{- 2 \bar{d} \bar{p} - \bar{r} - \bar{p} \bar{r}}{{\bar{E}}_{b} \bar{r}}] \bar{d} \bar{r} \sqrt{\frac{\bar{p}}{{\bar{E}}_{b} \bar{T}}} {\bar{E}}_{b}} . \end{aligned}

(21)

Finally, setting the LHS = g(y) and minimizing (21) yields the breakdown condition as

\begin{aligned} y_{0} & = \exp [\frac{- \bar{r} - \bar{p} \bar{r}}{{\bar{E}}_{b} \bar{r}}] \bar{d} \bar{r} \sqrt{\frac{\bar{p}}{{\bar{E}}_{b} \bar{T}}} {\bar{E}}_{b} (1 + 2 y_{0} {\bar{E}}_{b}) \exp [y_{0}] \\ - \exp [\frac{- 2 \bar{d} \bar{p} - \bar{r} - \bar{p} \bar{r}}{{\bar{E}}_{b} \bar{r}}] \bar{d} \bar{r} \sqrt{\frac{\bar{p}}{{\bar{E}}_{b} \bar{T}}} {\bar{E}}_{b} (1 + 2 y_{0} {\bar{E}}_{b}) \exp [y_{0}], \end{aligned}

(22)

where $y_{0} = (- 1 + \sqrt{1 + 8 {\bar{E}}_{b}}) / (4 {\bar{E}}_{b})$ ⁠. This is analogous to the final planar breakdown condition,²⁶ but accounts for the non-uniform electric field and resulting non-uniform space charge across the gap for a pin-to-plate geometry.

III. RESULTS

We first apply the breakdown model described in (22) to experimental data from Ref. 34 using a pin with a 0.5 μm tip radius and gap distances of 1, 5, and 10 μm. Figures 2–4 show the fitted $ϕ$ as a function of breakdown event for all data collected in Ref. 34 determined from numerically solving (22). Because we incorporate the work function into the scaling parameters, changing the work function changes all of the scaling terms in (15), which changes every subsequent variable (all of the dimensionless terms). Thus, care must be taken when comparing nondimensional values for different work functions. We numerically determine $ϕ$ that equates ${\bar{E}}_{b}$ from (22) with the experimentally determined, nondimensionalized breakdown voltage. Furthermore, (5) explicitly shows the dependence of j_FN on $ϕ$ ⁠.

FIG. 2.

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Work function as a function of breakdown event for 400 grit polishing at initial interelectrode gap distances (neglecting potential changes due to crater formation) of (a) 1 μm, (b) 5 μm, and (c) 10 μm. Each point represents data from each individual breakdown event from Ref. 34. Data from Brayfield et al., J. Appl. Phys. 125, 203302 (2019). Copyright 2019 AIP Publishing LLC.

FIG. 3.

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Work function as a function of breakdown event for 800 grit polishing at initial interelectrode gap distances (neglecting potential changes due to crater formation) of (a) 1 μm, (b) 5 μm, and (c) 10 μm. Each point represents a single breakdown event. Data from Brayfield et al., J. Appl. Phys. 125, 203302 (2019). Copyright 2019 AIP Publishing LLC.

FIG. 4.

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Work function as a function of breakdown event for 1200 grit polishing at initial interelectrode gap distances (neglecting potential changes due to crater formation) of (a) 1 μm, (b) 5 μm, and (c) 10 μm. Each point represents a single breakdown event. Data from Brayfield et al., J. Appl. Phys. 125, 203302 (2019). Copyright 2019 AIP Publishing LLC.

Next, we consider the data pertaining to the first and tenth breakdown events and the depths of the craters formed. Taking the sum of the initial interelectrode gap distance and the crater depth as d_eff, we fit $ϕ$ ⁠. Given the relative consistency of $ϕ$ over the wide range of $d_{e f f}$ ⁠, we averaged $ϕ$ and compared $V_{b}$ predicted by the average $ϕ$ with the experimental $V_{b}$ ⁠. Figure 5(b) shows that the predicted and measured $V_{b}$ differed by an average of ∼16%. We note that $ϕ$ is lower than typically stated in the literature (e.g., ∼4.5 eV for copper). We surmise that this behavior replaces a semiempirical correction factor that is usually accounted for by $β$ ⁠, which is now a purely geometrical effect; the role of the correction factor has shifted to $ϕ$ ⁠. Another possible contributory factor to the reduced $ϕ$ is that ions near the cathode may suppress the potential barrier, allowing the field-emitted electrons to tunnel into the gap which may lower the work function of the cathode. While some studies^44,45 have suggested that space charge could lower $ϕ$ ⁠, further investigation is required. For instance, Malayter and Garner theoretically investigated the effect of surface waviness on measured $ϕ$ using a scanning Kelvin probe (SKP); they calculated the change in the charge in the gap, and ultimately the capacitance, of each step of the SKP.³⁹ It may be possible to modify this theory to account for the change in $ϕ$ as space charge builds up in the gap.

FIG. 5.

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(a) Work function as a function of effective gap distance d_eff (interelectrode gap distance + crater depth) based on fitting the experimental data³⁴ before and after crater formation. Each data point is an individual experiment. (b) Measured³⁴ and predicted breakdown voltage using the average work function as a function of d_eff.

Figure 6 uses (22) and the appropriate scaling parameters to predict breakdown voltage as a function of work function for various pressures. Increasing the work function increases the breakdown voltage, since increasing the work function increases the energy required for electrons to overcome the potential barrier and tunnel through the cathode. Figure 6 also suggests that pressure does not have a significant effect on the breakdown voltage in (a)–(c), but increases the breakdown voltage for (d) at the largest work function. We will explicitly investigate gap distance and pressure next.

FIG. 6.

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Breakdown voltage as a function of work function for various gap distances at pressures of (a) 1000 Torr, (b) 760 Torr, (c) 380 Torr, and (d) 50 Torr.

Next, we consider the nondimensionalized version of the model. While our most recent nondimensionalized breakdown models^23,38 considered work function as a separate dimensionless variable and derived corresponding scaling parameters, the present study keeps the work function within the scaling parameters as in our earlier studies.³⁶ Thus, the scaling parameters for each gas in the present study are only constant for a given work function. Figure 7 shows the dimensionless breakdown voltage as a function of dimensionless gap distance for various dimensionless pressures and work functions. Figure 8 shows the dimensionless breakdown voltage as a function of dimensionless pressure for various dimensionless gap distances and work functions. Overall, these results indicate that breakdown voltage depends more strongly on gap distance than pressure in the field emission driven regime, in agreement with prior sensitivity analyses of field emission breakdown theories for a parallel plate geometry.⁴³

FIG. 7.

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Dimensionless breakdown voltage $\bar{V}$ as a function of dimensionless gap distance $\bar{d}$ for various dimensionless pressures $\bar{p}$ for work function $ϕ$ of (a) 1 eV, (b) 2 eV, and (c) 3 eV. Because changing the work function $ϕ$ changes the scaling parameters, the dimensionless values in (a)–(c) should not be compared to one another.

FIG. 8.

View large Download slide

Dimensionless breakdown voltage $\bar{V}$ as a function of dimensionless pressure $\bar{p}$ for various dimensionless gap distances $\bar{d}$ for work function $ϕ$ of (a) 1 eV, (b) 2 eV, and (c) 3 eV. Because changing the work function $ϕ$ changes the scaling parameters, the dimensionless values in (a)–(c) should not be compared to one another.

Furthermore, we can assess the impact of tip radius on breakdown voltage. Figure 9(a) shows the breakdown voltage as a function of gap distance for tip radii of 0.5, 1, and 3 μm at a pressure of 760 Torr and work function of 2 eV, Fig. 9(b) shows the dependence on the ratio of gap distance to tip radius, d/r, for the same conditions, and Fig. 9(c) shows breakdown voltage as a function of tip radius for gap distances of 0.5 and 2 μm. As demonstrated, the sharpest tip corresponds to the lowest breakdown voltage, and increasing tip radius for a constant gap distance shows the breakdown voltage approaching a constant, as expected. As d/r becomes small, the geometry begins to more closely resemble a parallel plate geometry, so the breakdown voltage becomes less sensitive to r.

FIG. 9.

View large Download slide

(a) Breakdown voltage as a function of gap distance for tip radii of 0.5, 1, and 3 μm. (b) Breakdown voltage as a function of the ratio of gap distance to tip radius d/r for tip radii of 0.5, 1, and 3 μm. (c) Breakdown voltage as a function of tip radius for gap distances of 0.5 and 2 μm.

As discussed above, this model eliminates the dependence on the secondary emission coefficient $γ_{S E}$ in the field emission driven breakdown regime after (17) due to the series expansion, similar to our initial microscale gas breakdown study.³⁶ This assumption begins to weaken as we approach the PL regime. We can identify the condition for transitioning from this simplified field emission driven regime theory to the traditional PL by considering

1 - γ_{S E} \int_{0}^{\bar{d}} \bar{α} (\bar{x}) d \bar{x} = 0,

(23)

which is a dimensionless, position-dependent modified Townsend avalanche condition. Inserting our definitions from above and solving for γ_SE yields

\begin{aligned} γ_{S E} & = - \frac{\ln (2 \bar{d} / \bar{r} + 1)}{2 \bar{V}} [\exp (- \frac{\ln (2 \bar{d} / \bar{r} + 1) \bar{p} (2 \bar{d} + \bar{r})}{2 \bar{V}}) \\ {- \exp (- \frac{\bar{p} \bar{r} \ln (2 \bar{d} / \bar{r} + 1)}{2 \bar{V}})]}^{- 1}, \end{aligned}

(24)

which determines γ_SE to transition to the traditional PL. Figure 10 plots this transition value of $γ_{S E}$ for the parameters considered in Fig. 7. For all but the largest $\bar{p}$ and $\bar{d}$ considered, achieving the transition to the classical PL requires γ_SE much larger than typical magnitudes. Thus, assuming that γ_SE minimally impacts the breakdown voltage calculation is reasonable for the majority of the parameter space considered here. For reference, the dimensional pressures and gap distances considered here correspond to 50–1000 Torr and 0.1–5 μm, respectively.

FIG. 10.

View large Download slide

Secondary emission coefficient γ_SE as a function of dimensionless gap distance $\bar{d}$ for various pressures $\bar{p}$ at work function $ϕ$ of (a) 1 eV, (b) 2 eV, and (c) 3 eV.

IV. CONCLUSION

This work derived a dimensionless breakdown model for a pin-to-plate geometry considering a non-uniform electric field, field emission, and Townsend effects. We compared the theory to experimental data to determine work function values to assess variation of the theoretical predictions over the course of several breakdown events. Additionally, we showed the variation of breakdown voltage with both pressure and gap distance, demonstrating a stronger dependence on gap distance, as expected based on previous analyses for planar diodes.⁴³

The lower work function obtained by fitting $ϕ$ to experimental data rather than $β$ may be due to the presence of additional space charge lowering the energy needed for electrons to tunnel into the gap or an artifact of residual effects from the semiempirical correction factor usually accounted for by fitting $β$ ⁠. Further investigation is needed to elucidate the physics behind the ∼2–3 eV difference in work function compared to typically reported values from the literature.

Additionally, further experimental data assessing crater depth is needed to distinguish changes in breakdown voltage due to surface morphology effects on work function and crater depth induced changes on modified gap distance. Future work will further examine the effects of electrode geometry and the resulting charge buildup on work function and the application of this technique to other relevant geometries, such as sphere-to-sphere.^31,32,46

ACKNOWLEDGMENTS

This material is based on work supported by the Air Force Office of Scientific Research under Award No. FA9550-18-1-0218 and the Office of Naval Research under Grant No. N00014-17-1-2702. A.M.L. gratefully acknowledges funding from a graduate scholarship from the Directed Energy Professional Society.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

REFERENCES

1.

S.

Li

,

Y.

Huang

,

D.

Min

,

G.

Qu

,

H.

Niu

,

Z.

Li

,

W.

Wang

,

J.

Li

, and

W.

Liu

, “

Synergic effect of adsorbed gas and charging on surface flashover

,”

Sci. Rep.

9

,

5464

(

2019

).

https://doi.org/10.1038/s41598-019-41961-0

Google Scholar

Crossref

PubMed

2.

A. L.

Garner

,

A.

Caiafa

,

Y.

Jiang

,

S.

Klopman

,

C.

Morton

,

A. S.

Torres

,

A. M.

Loveless

, and

V. B.

Neculaes

, “

Design, characterization and experimental validation of a compact, flexible pulsed power architecture for ex vivo platelet activation

,”

PLoS One

12

,

e0181214

(

2017

).

https://doi.org/10.1371/journal.pone.0181214

Google Scholar

Crossref

PubMed

3.

W.

Zhong

,

G.

Zhang

, and

A.

Xu

, “

Detachment of microparticles from electrode erosion and the subsequent breakdown

,”

AIP Adv.

9

,

045023

(

2019

).

https://doi.org/10.1063/1.5085825

Google Scholar

Crossref

4.

A.

Darwish

,

S. S.

Refaat

,

H. A.

Toliyat

, and

H.

Abu-Rub

, “

On the electromagnetic wave behavior due to partial discharge in gas insulated switchgears: State-of-art review

,”

IEEE Access

7

,

75822

–

75836

(

2019

).

https://doi.org/10.1109/ACCESS.2019.2921089

Google Scholar

Crossref

5.

P.

Zhao

and

L.

Guo

, “

Effect of air breakdown in near-field region on maximum power radiated from aperture antenna

,”

J. Electromagn. Waves Appl.

30

,

795

–

804

(

2016

).

https://doi.org/10.1080/09205071.2016.1154484

Google Scholar

Crossref

6.

S.

Yan

,

A. D.

Greenwood

, and

J.-M.

Jin

, “

Simulation of high-power microwave air breakdown modeled by a coupled Maxwell–Euler system with a non-Maxwellian EEDF

,”

IEEE Trans. Antennas Propag.

66

,

1882

–

1893

(

2018

).

https://doi.org/10.1109/TAP.2018.2804482

Google Scholar

Crossref

7.

D. R.

Welch

,

M. E.

Cuneo

,

C. L.

Olson

, and

T. A.

Mehlhorn

, “

Gas breakdown effects in the generation and transport of light ion beams for fusion

,”

Phys. Plasmas

3

,

2113

–

2121

(

1996

).

https://doi.org/10.1063/1.871663

Google Scholar

Crossref

8.

X.

Song

,

X. R.

Duan

,

X. M.

Song

,

G. Y.

Zheng

,

S.

Wang

,

B.

Li

,

X. Y.

Bai

,

S. D.

Song

,

C.

Wang

, and

J.

Sun

, “

Experimental results of plasma breakdown and flux optimization on HL-2A tokamak

,”

Fusion Eng. Des.

125

,

195

–

198

(

2017

).

https://doi.org/10.1016/j.fusengdes.2017.11.007

Google Scholar

Crossref

9.

K. C.

Hammond

,

R.

Raman

, and

F. A.

Volpe

, “

Application of Townsend avalanche theory to tokamak startup by coaxial helicity injection

,”

Nucl. Fusion

58

,

016013

(

2018

).

https://doi.org/10.1088/1741-4326/aa8fa4

Google Scholar

Crossref

10.

B.

Lloyd

,

G. L.

Jackson

,

T. S.

Taylor

,

E. A.

Lazarus

,

T. C.

Luce

, and

R.

Prater

, “

Low voltage ohmic and electron cyclotron heating assisted startup in DIII-D

,”

Nucl. Fusion

31

,

2031

–

2053

(

1991

).

https://doi.org/10.1088/0029-5515/31/11/001

Google Scholar

Crossref

11.

Y.

Peng

,

W.

Jiang

,

M. E.

Innocenti

,

Y.

Zhang

,

X.

Hu

,

G.

Zhuang

, and

G.

Lapenta

, “

On the breakdown modes and parameter space of ohmic tokamak start-up

,”

J. Plasma Phys.

84

,

905840505

(

2018

).

https://doi.org/10.1017/S0022377818001046

Google Scholar

Crossref

12.

K. A.

Rezvykh

and

V. A.

Romanov

, “

Gases breakdown voltage calculation for the case of accelerator nonuniform fields by the method of base

,”

Nucl. Instrum. Meth. Phys. Res. Sect. A

423

,

203

–

212

(

1999

).

https://doi.org/10.1016/S0168-9002(98)00913-9

Google Scholar

Crossref

13.

A. M. R. N.

Alrashidi

,

N. M.

Adam

,

A. A.

Hairuddin

, and

L. C.

Abdullah

, “

A review on plasma combustion of fuel in internal combustion engines

,”

Int. J. Energy Res.

42

,

1813

–

1833

(

2018

).

https://doi.org/10.1002/er.3964

Google Scholar

Crossref

14.

J.

Kolbeck

,

A.

Anders

,

I. I.

Beilis

, and

M.

Keidar

, “

Micro-propulsion based on vacuum arcs

,”

J. Appl. Phys.

125

,

220902

(

2019

).

https://doi.org/10.1063/1.5081096

Google Scholar

Crossref

15.

D.

Lev

,

R. M.

Myers

,

K. M.

Lemmer

,

J.

Kolbeck

,

H.

Koizumi

, and

K.

Polzin

, “

The technological and commercial expansion of electric propulsion

,”

Acta Astronautica

159

,

213

–

227

(

2019

).

https://doi.org/10.1016/j.actaastro.2019.03.058

Google Scholar

Crossref

16.

Y.

Ju

and

W.

Sun

, “

Plasma assisted combustion: Dynamics and chemistry

,”

Prog. Energy Combust. Sci.

48

,

21

–

83

(

2015

).

https://doi.org/10.1016/j.pecs.2014.12.002

Google Scholar

Crossref

17.

K. H.

Becker

,

K. H.

Schoenbach

, and

J. G.

Eden

, “

Microplasmas and applications

,”

J. Phys. D: Appl. Phys.

39

,

R55

–

R70

(

2006

).

https://doi.org/10.1088/0022-3727/39/3/R01

Google Scholar

Crossref

18.

K. H.

Schoenbach

and

K.

Becker

, “

20 years of microplasma research: A status report

,”

Eur. Phys. J. D

70

(

2

),

29

(

2016

).

https://doi.org/10.1140/epjd/e2015-60618-1

Google Scholar

Crossref

19.

K.

Becker

,

A.

Koutsospyros

,

S.-M.

Yin

,

C.

Christodoulatos

,

N.

Abramzon

,

J. C.

Joaquin

, and

G.

Brelles-Mariño

, “

Environmental and biological applications of microplasmas

,”

Plasma Phys. Controlled Fusion

47

,

B513

–

B523

(

2005

).

https://doi.org/10.1088/0741-3335/47/12B/S37

Google Scholar

Crossref

20.

M. G.

Kong

,

G.

Kroesen

,

G.

Morfill

,

T.

Nosenko

,

T.

Shimizu

,

J.

van Dijk

, and

J. L.

Zimmerman

, “

Plasma medicine: An introductory review

,”

New J. Phys.

11

,

115012

(

2009

).

https://doi.org/10.1088/1367-2630/11/11/115012

Google Scholar

Crossref

21.

M.

Laroussi

, “

Low-temperature plasma jet for biomedical applications: A review

,”

IEEE Trans. Plasma Sci.

43

,

703

–

712

(

2015

).

https://doi.org/10.1109/TPS.2015.2403307

Google Scholar

Crossref

22.

F.

Paschen

, “

Ueber die zum funkenübergang in luft, wasserstoff und kohlensäure bei verschiedenen drucken erforderliche potentialdifferenz

,”

Ann. Phys.

273

,

69

–

96

(

1889

).

https://doi.org/10.1002/andp.18892730505

Google Scholar

Crossref

23.

A. L.

Garner

,

A. M.

Loveless

,

J. N.

Dahal

, and

A.

Venkattraman

, “

A tutorial on theoretical and computational techniques for gas breakdown in microscale gaps

,”

IEEE Trans. Plasma Sci.

48

,

808

–

824

(

2020

).

https://doi.org/10.1109/TPS.2020.2979707

Google Scholar

Crossref

24.

R. H.

Fowler

and

L.

Nordheim

, “

Electron emission in intense electric fields

,”

Proc. R. Soc. London A

119

,

173

–

181

(

1928

).

https://doi.org/10.1098/rspa.1928.0091

Google Scholar

Crossref

25.

W. S.

Boyle

and

P.

Kisliuk

, “

Departure from Paschen’s law of breakdown in gases

,”

Phys. Rev.

97

,

255

–

259

(

1955

).

https://doi.org/10.1103/PhysRev.97.255

Google Scholar

Crossref

26.

A.

Venkattraman

and

A. A.

Alexeenko

, “

Scaling law for direct current field emission-driven microscale gas breakdown

,”

Phys. Plasmas

19

,

123515

(

2012

).

https://doi.org/10.1063/1.4773399

Google Scholar

Crossref

27.

D. B.

Go

and

A.

Venkattraman

, “

Microscale gas breakdown: Ion enhanced field emission and the modified Paschen’s curve

,”

J. Phys. D: Appl. Phys.

47

,

503001

(

2014

).

https://doi.org/10.1088/0022-3727/47/50/503001

Google Scholar

Crossref

28.

M.

Radmilović-Radjenović

and

B.

Radjenović

, “

Theoretical study of the electron field emission phenomena in the generation of a micrometer scale discharge

,”

Plasma Sources Sci. Technol.

17

,

024005

(

2008

).

https://doi.org/10.1088/0963-0252/17/2/024005

Google Scholar

Crossref

29.

M.

Radmilović-Radjenović

,

Š.

Matejčik

,

M.

Klas

, and

B.

Radjenović

, “

The role of the field emission effect in direct-current argon discharges for the gaps ranging from 1 to 100 μm

,”

J. Phys. D: Appl. Phys.

46

,

015302

(

2013

).

https://doi.org/10.1088/0022-3727/46/1/015302

Google Scholar

Crossref

30.

Y.

Fu

,

P.

Zhang

,

J. P.

Verboncoeur

, and

X.

Wang

, “

Electrical breakdown from macro to micro/nano scales: A tutorial and a review of the state of the art

,”

Plasma Res. Express

2

,

013001

(

2020

).

https://doi.org/10.1088/2516-1067/ab6c84

Google Scholar

Crossref

31.

G.

Meng

,

X.

Gao

,

A. M.

Loveless

,

C.

Dong

,

D.

Zhang

,

K.

Wang

,

B.

Zhu

,

Y.

Chen

, and

A. L.

Garner

, “

Demonstration of field emission driven microscale gas breakdown for pulsed voltages using in-situ optical imaging

,”

Phys. Plasmas

25

,

082116

(

2018

).

https://doi.org/10.1063/1.5046335

Google Scholar

Crossref

32.

G.

Meng

,

Q.

Ying

,

A. M.

Loveless

,

F.

Wu

,

K.

Wang

,

Y.

Fu

,

A. L.

Garner

, and

Y.

Cheng

, “

Spatio-temporal dynamics of pulsed gas breakdown in microgaps

,”

Phys. Plasmas

26

,

014506

(

2019

).

https://doi.org/10.1063/1.5081009

Google Scholar

Crossref

33.

M. A.

Bilici

,

J. R.

Haase

,

C. R.

Boyle

,

D. B.

Go

, and

R. M.

Sankaran

, “

The smooth transition from field emission to a self-sustained plasma in microscale electrode gaps at atmospheric pressure

,”

J. Appl. Phys.

119

,

223301

(

2016

).

https://doi.org/10.1063/1.4953648

Google Scholar

Crossref

34.

R. S.

Brayfield

II,

A. J.

Fairbanks

,

A. M.

Loveless

,

S.

Gao

,

A.

Dhanabal

,

W.

Li

,

C.

Darr

,

W.

Wu

, and

A. L.

Garner

, “

The impact of cathode surface roughness and multiple breakdown events on microscale gas breakdown at atmospheric pressure

,”

J. Appl. Phys.

125

,

203302

(

2019

).

https://doi.org/10.1063/1.5085217

Google Scholar

Crossref

35.

Y.

Fu

,

J.

Krek

,

P.

Zhang

, and

J. P.

Verboncoeur

, “

Evaluating microgap breakdown mode transition with electric field non-uniformity

,”

Plasma Sources Sci. Technol.

27

,

095014

(

2018

).

https://doi.org/10.1088/1361-6595/aadf56

Google Scholar

Crossref

36.

A. M.

Loveless

and

A. L.

Garner

, “

Scaling laws for gas breakdown for nanoscale to microscale gaps at atmospheric pressure

,”

Appl. Phys. Lett.

108

,

234103

(

2016

).

https://doi.org/10.1063/1.4953202

Google Scholar

Crossref

37.

A. M.

Loveless

and

A. L.

Garner

, “

Generalization of microdischarge scaling laws for all gases at atmospheric pressure

,”

IEEE Trans. Plasma Sci.

45

,

574

–

583

(

2017

).

https://doi.org/10.1109/TPS.2017.2647988

Google Scholar

Crossref

38.

A. M.

Loveless

and

A. L.

Garner

, “

A universal theory for gas breakdown from microscale to the classical Paschen law

,”

Phys. Plasmas

24

,

113522

(

2017

).

https://doi.org/10.1063/1.5004654

Google Scholar

Crossref

39.

J. R.

Malayter

and

A. L.

Garner

, “

Theoretical assessment of surface waviness on work function

,”

AIP Adv.

10

,

095110

(

2020

).

https://doi.org/10.1063/5.0016116

Google Scholar

Crossref

40.

E. L.

Murphy

and

R. H.

Good

, “

Thermionic emission, field emission, and the transition region

,”

Phys. Rev.

102

,

1464

–

1473

(

1956

).

https://doi.org/10.1103/PhysRev.102.1464

Google Scholar

Crossref

41.

D.

Mariotti

,

J. A.

McLaughlin

, and

P.

Maguire

, “

Experimental study of breakdown voltage and effective secondary electron emission coefficient for a micro-plasma device

,”

Plasma Sources Sci. Technol.

13

,

207

–

212

(

2004

).

https://doi.org/10.1088/0963-0252/13/2/003

Google Scholar

Crossref

42.

D.

Marić

,

M.

Savić

,

J.

Sivoš

,

N.

Škoro

,

M.

Radmilović-Radjenović

,

G.

Malović

, and

Z. L.

Petrović

, “

Gas breakdown and secondary electron yields

,”

Eur. Phys. J. D

68

,

155

(

2014

).

https://doi.org/10.1140/epjd/e2014-50090-x

Google Scholar

Crossref

43.

S. D.

Dynako

,

A. M.

Loveless

, and

A. L.

Garner

, “

Sensitivity of modeled microscale gas breakdown voltage due to parametric variation

,”

Phys. Plasmas

25

,

103505

(

2018

).

https://doi.org/10.1063/1.5042270

Google Scholar

Crossref

44.

Y.

Li

and

D. B.

Go

, “

The quantum mechanics of ion-enhanced field emission and how it influences microscale gas breakdown

,”

J. Appl. Phys.

116

,

103306

(

2014

).

https://doi.org/10.1063/1.4895634

Google Scholar

Crossref

45.

J. P.

Barbour

,

W. W.

Dolan

,

J. K.

Trolan

,

E. E.

Martin

, and

W. P.

Dyke

, “

Space-charge effects in field emission

,”

Phys. Rev.

92

,

45

–

51

(

1953

).

https://doi.org/10.1103/PhysRev.92.45

Google Scholar

Crossref

46.

A. M.

Loveless

,

G.

Meng

,

Q.

Ying

,

F.

Wu

,

K.

Wang

,

Y.

Cheng

, and

A. L.

Garner

, “

The transition to Paschen’s law for microscale gas breakdown at subatmospheric pressure

,”

Sci. Rep.

9

,

5669

(

2019

).

https://doi.org/10.1038/s41598-019-42111-2

Google Scholar

Crossref

PubMed

2021

Author(s)

Analytic theory for field emission driven microscale gas breakdown for a pin-to-plate geometry

I. INTRODUCTION

II. MODEL

III. RESULTS

IV. CONCLUSION

ACKNOWLEDGMENTS

DATA AVAILABILITY

REFERENCES

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Analytic theory for field emission driven microscale gas breakdown for a pin-to-plate geometry Free

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Analytic theory for field emission driven microscale gas breakdown for a pin-to-plate geometry