Plasma Sources Science and Technology Plasma Sources Sci. Technol. 27 (2018) 095014 (8pp) https://doi.org/10.1088/1361-6595/aadf56 Evaluating microgap breakdown mode transition with electric field non-uniformity Yangyang Fu1,2 , Janez Krek1 , Peng Zhang2 and John P Verboncoeur1,2 1 Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, United States of America 2 Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, United States of America E-mail: [email protected] Received 29 July 2018, revised 22 August 2018 Accepted for publication 6 September 2018 Published 25 September 2018 Abstract This paper evaluates the transitions between the breakdown modes that are dominated by field emission (FE) and by secondary electron emission (SEE) in argon microgaps, based on a mathematical model originally proposed for the modified Paschen’s curve (Go and Pohlman 2010 J. Appl. Phys. 107 103303). In the present model, instead of assuming a constant electric field across the gap, the spatial distribution of the electric field due to the presence of a parabolic cathode protrusion is implemented, and a modification of the first Townsend ionization coefficient for microscale gaps from particle-in-cell/Monte Carlo collision simulations is employed (Venkattraman and Alexeenko 2012 Phys. Plasmas 19 123515). The breakdown mode transitions are quantified by the variations of gas pressure and gap distance. It is found that assuming a constant electric field across the gap will overestimate the electron avalanche, which underestimates the breakdown voltage and shifts the breakdown mode transitions to a smaller gap. By varying the field enhancement factor, the critical point of the breakdown mode transition can be significantly adjusted. Increasing the field enhancement factor will lower the breakdown voltage in the FE regime and increase the breakdown voltage in the SEE regime. Keywords: gas breakdown, Paschen’s law, microdischarge, field emission, Townsend discharge, secondary electron emission, breakdown mode transition 1. Introduction combined with the SEE and the FE results in microgap breakdown in different regimes. The gas breakdown mechanisms in microgaps (less than Deviations from Paschen’s law of gas breakdown in the 10 μm) transiting from the field emission (FE) dominant presence of field emission were observed in the 1950s regime to the secondary electron emission (SEE) dominant [14–17]. To elucidate the breakdown deviations with field regime are widely reported both experimentally and numeri- emission, an empirical form for the electron current density cally [1–5]. Understanding these phenomena is of critical driven by the cathode electric field was proposed by Boyle importance for many prospective applications in nano- and and Kisliuk [15]. Based on this empirical form, Ramilović- micro-scale electronics, including micro-electro-mechanical Radjenović and Radjenović derived a mathematical model for systems, micro-switches, and microchip devices [6–10].As microgap breakdown caused by ion-enhanced field emission, previously reported, the Townsend theory for micro- which agrees with the particle-in-cell/Monte Carlo collision discharges holds until field emission becomes dominant (PIC/MCC) model when the discharge is in the field emission [11–13]. The key physical mechanisms for the breakdown regime [18–20]. However, as Ramilović-Radjenović and processes are the electron avalanche process (the α process) Radjenović mentioned in [19], this model will overestimate across the gap and the surface emission processes (the γ the breakdown voltage as the gap distance increases where the process, such as SEE and FE) at the cathode. The α process field emission gradually disappears and the SEE becomes 0963-0252/18/095014+08$33.00 1 © 2018 IOP Publishing Ltd Plasma Sources Sci. Technol. 27 (2018) 095014 YFuet al dominant. Later in [21, 22], Go, Pohlman, and Tirumala widely used and assumes that the electrons undergo only proposed a formulation of the so-called modified Paschen’s ionizing collisions, which may be realistic at high reduced curve, combining the fieldemissionandtheSEEcoefficients electric fields and moderate energies; equation (1B) is into the Townsend breakdown criteria, which successfully usually used for inert gas since it gives a better fittothe describes the breakdown mode transition between the field experimental data [32]. For the gap with uniform electric emission regime and the SEE regime. In that model, the field, the Townsend avalanche criteria for gas breakdown is Fowler–Nordheim (F–N) equation is employed to describe expressed as the field emission current density and the impact of the ad uncertainty of the related parameters, such as work function, gSEE ()e11,-= () 2 field enhancement factor, and other fitting parameters, on the fi breakdown curves has been assessed [22–24]. One limitation where gSEE is the SEE coef cient that depends on both of the model is that the electric field is treated as a constant cathode material and the gas. Combining equations (1A) and across the gap and it is suddenly enhanced at the cathode by (2) yields the conventional Paschen’scurve an assumed factor. Though making the formula simple and fi Bpd generally clear, the assumption of constant electric eld is Ub = ⎡ ⎤ .3() physically less appropriate and often violated, which might ln()pd + ln⎢ A ⎥ ⎣ ln() 1g + 1 ⎦ result in unexpected misbehaviors in calculating the break- SEE fi down voltages. The effect of the electric eld non-uni- Equation (3) describes the breakdown voltage when ’ formity on Paschen s curve in macrogaps can be found in cathode emission only includes SEE [31, 32].Whenthe [ – ] 25 29 . microgap size becomes smaller than several microns (or fi In this work, the non-uniformity of the electric eld equivalently, the electric field strength is on the order of across the gap due to the presence of a parabolic cathode tip is 109 Vm−1), field emission can also contribute to the fi considered for calculations of the cathode eld emission and cathode emission. The FE current density j described by fi fi FN the rst Townsend ionization coef cient. The contributions the F–N equation is from the field emission and the SEE on the electron avalanche 22 ⎡ 32 ⎤ are evaluated in argon microgap breakdowns. The breakdown AEFN b ByFN ju() fi j =-exp⎢ ⎥ ,() 4 mode transitions are quanti ed by the variations of the gas FN jty2 () ⎣ bE ⎦ pressure and the gap distance. In section 2, the mathematical model for microgap breakdown is introduced and the treat- where A and B are the F–Nconstants,E is the applied fi fi FN FN ment of the electric eld and the rst Townsend ionization electric field defined as E = Ud, j is the work function of fi ’ coef cient is discussed. In section 3, Paschen s curves cal- the cathode, and β is the local field enhancement factor. The fi culated using different ionization coef cients are compared, parameter y is a function of j, β,andE,andthebarrier followed by the evaluation of the transition characteristics of shape function ty2 ()and u()y have established approxima- the discharge from the FE regime to the SEE regime. The tions [33, 34].Usingthisfield emission expression, the effects of the non-uniform electric field and the field effective field emission coefficient is given by enhancement factor are also discussed. Concluding remarks are given in section 4. ⎛ ⎞ ⎜⎟DFN gFE =-K exp ,() 5 ⎝ E ⎠ 2. Mathematical model where K and DFN are material and gas dependent constants. In Townsend gas breakdowns, the α process leads to electron The constant K is related to the ion enhancement effect and fi impact ionization across the gap and the γ process results in can be determined from the ratio of the eld emission electron emissions from the cathode, both of which are current density to the positive ion current density into the [ ] important for self-sustained discharges [30–32]. Depending cathode 19 . However, the experimental determination may fi on the discharge conditions, the cathode emission could be be quite dif cult since there are complications due to [ – ] either ion-enhanced field emission or ion-induced SEE. The electron attachment, ionization by metastables, etc 18 20 . first Townsend ionization coefficient α is used to describe In this study, K is treated as a constant much larger than 1 (∼ 7) fi [ ] the electron-impact ionization during avalanches, which is 10 and DFN is de ned as 15, 21 given by 32 9 j a =-Apexp() Bpd U , () 1 A DFN =´()6.85 10 . () 6 b or The modified breakdown criteria, which includes FE-driven a =-Cpexp() D pd U , () 1 B and SEE-driven breakdown, is given by Go and Pohlman as [ ] where A, B, C,andD are constants based on the gas type, p follows 21 is the gas pressure, d is the gap distance, and U is the ad applied voltage in the gap [21]. Note that equation (1A) is ()()ggSEE+-= FE e11. () 7 2 Plasma Sources Sci. Technol. 27 (2018) 095014 YFuet al comparable to or smaller than the length scale over which the electric field changes. In order to consider the effect of the non-uniformity of the electric field, a parabolic surface protrusion as the cathode facing an infinite plane electrode is employed, as shown in figure 1(a), where x is the distance from the protrusion tip along the axis, d is the gap distance, r is the curvature radius, and R is the anode radius (R d). The electric field dis- tribution along the axis is given by [32] 2U Ex()= .8() ()()2ln21xr++ dr For a given gap distance d, an expected field enhance- ment factor β at the cathode can be achieved by adjusting the geometry parameter r in equation (8), i.e., b = (2ln21.dr)( dr+ )Note that in this model, the field enhancement factor is only due to the geometrical effects and works as an effective global field enhancement factor.