A MATHEMATICAL MODEL FOR THE DEPARTURE FROM PASCHEN S LAW AT MICROMETER GAPS USING ION ENHANCED FIELD EMISSION

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1 AME6637: Ionization and Ion Transport Final Project Paper May 5, 1 Notre Dame, IN USA A MATHEMATICA MODE FOR THE DEPARTURE FROM PASCHEN S AW AT MICROMETER GAPS USING ION ENHANCED FIED EMISSION Rakshit Tirumala Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, IN ABSTRACT Paschen s curve describes the relationship between the breakdown potential between two electrodes and the distance between them at a fixed pressure. Experimental evidence points out the fact that the euation describing this relationship is not valid at very small gap distances (< 1 µm) and that another phenomenon, namely field emission plays a prominent role in these regimes. Boyle and Kisliuk [6] proposed a theory in this regime based on ionenhanced field emission that was later extended by Ramilovic-Radjenovic and Radjenovic [1] to provide a formulation for the breakdown potential in these small gaps. However, the formulation provided has an undetermined parameter that has been assumed to be much larger than 1 (~1 7 ). Go and Pohlman [11] developed a model combining the Paschen s curve and the formulation proposed in [1] to obtain a relationship that describes the transition from the small scale field emission regime to the large scale regime described by Paschen s euation. However, in all these models, the empirical parameter is adjusted to fit data. Also, the assumptions made in [6] are shown here to be incorrect. The present work proposes a mathematical model to predict the breakdown potential based only on gas constants and material properties which are relatively well established. The model proposed is based on ion-enhanced field emission as in [6] but studies the more fundamental nature of the phenomenon to provide a complete description. INTRODUCTION The advancing miniaturization of electrical components propels the study of electrical breakdown in micrometer gaps. Higher conductor density in switches, connectors and micro-electromechanical systems (MEMS) has the inter-electrode spacing being reduced to micrometer gaps and lower. At these small gaps, even a small potential difference can cause the breakdown of the interstitial gas. Thus electrical breakdown of gaseous dielectrics becomes an important failure mode in micro-scale regimes. Additionally, in recent years, microplasmas are being extensively studied for use in various environmental applications such as for the reduction for NOx emissions and destruction of volatile organic compounds (VOC), as UV radiation sources, for gas sesnsing and plasma reactors [1,]. As such, the proper design of the electrical system reuires the understanding of the electrical behavior at microgaps. The primary focus of this paper is to predict the potential at which a uniform field gas discharge transitions from a non self-sustaining regime to a selfsustaining regime. This transition, called a Townsend breakdown, is a volumetric mode of breakdown characteristic of uniform fields at moderate gas pressures. The other form of breakdown is called a streamer breakdown and is common in high pressure gases. Townsend s breakdown criterion describes the condition at which the electron impact ionization processes in the gas (α-processes) and the secondary electron emission processes at the cathode surface (γprocesses) are critically matched so as to result in an electron avalanche within the gas. Traditionally, Paschen s euation, derived from the Townsend s criterion, expresses the breakdown potential (V b ) as a function of the inter-electrode gap distance (d), the gas pressure (p) and the gas composition [3]. However, this euation is known to fail at small interelectrode gaps (d < 7 µm) and at high pressures. Micro-scale breakdown has been studied for a few decades going back to the experimental observations of this deviation from Paschen s law at 1.1

2 small gaps [4,5]. These experimental observations as well as deviations at high pressures prompted a series of theoretical and experimental studies. In 1955, Boyle and Kisliuk [6] put forth a theory based on the enhancement of the cathodic field emission due to the presence of positive ions near the surface. This phenomenon, known as ion-enhanced field emission, was shown to cause a significant contribution to the electron production at small gaps resulting in a smaller breakdown potential. In the past couple of decades, various other studies have focused on micrometer breakdown regimes in air at atmospheric pressure [7-9] experimentally, theoretically as well as computationally. In the past few years, studies by Radmilovich- Radjenovic and Radjenovic [1] have continued the work of Boyle and Kisliuk [6] in formulating an expression for the breakdown potential using the emission ion coefficient (γ ) for the ion-enhanced field emission. However, this formulation is only valid at small gaps where this phenomenon dominates and has been shown by Go and Pohlman [11] to increasingly deviate as the gap distance is increased beyond 5 µm. Go and Pohlman [11], taking into account the secondary emission ion coefficient (γ i ) along with ion-enhanced field emission formulate a more complete theoretical expression for the breakdown potential that is shown to be valid over the entire range. Experiments have typically shown three broad regimes in a atmospheric pressure breakdown curve when plotted against the gap distance an increasing regime at small gaps (< ~5 µm), a plateau region with constant breakdown potential (~ 5-8 µm) and a region that adheres to Paschen s euation for large gaps (> ~8 µm). The formulation by Go and Pohlman has been shown to describe these three regimes clearly. However, as will be shown in the following sections, the fundamental theory by Boyle and Kisliuk [6] on which the later studies are based makes a set of assumptions which are either inaccurate or have at best an empirical basis. As such, all the later formulations based on this theory incorporate an empirical parameter (K) that has been assumed by the later studies to be much larger than 1 (~1 7 ) when extracting the breakdown potential from their respective formulations. The parameter however, being based on multiple empirical assumptions, cannot be obtained directly from experiments. The only way to experimentally determine the parameter K is to obtain the breakdown potentials, hence rendering any theoretical formulation effectively incomplete. The present model of ion-enhanced field emission, based on the work by Kisliuk in 1959 [1], and its effect on the breakdown potential eliminates the above parameter K and expresses the breakdown potential as a transcendental euation in terms of the inter-electrode distance (d), gas composition, and cathode material and surface properties. ITERATURE REVIEW Basic Mechanisms: Townsend s breakdown criteria is a critical condition relating the two processes of electron impact ionization collisions in the gas α-processes) and the secondary electron emission processes at the cathode surface (γ-processes) such that the net current produced at the cathode is infinite. The condition is given by [3] γ (1) ( 1) 1 i e αd where γ i (~.1) is the secondary emission coefficient due to ion impact on the cathode surface, d is the inter-electrode gap distance and α is the Townsend s first ionization coefficient which describes the generation of gas ions by electron impact. In uniform fields, α has an empirical formulation given by, Bpd Ape V α () where A and B are experimentally determined gas constants, V is the applied potential and p is the interstitial gas pressure. γ is the ratio which describes the yield of electrons at the cathode due to the impacting ions. Thus n n e, emitted i, impact j e, emitted γ (3) j i, impact Substituing E. into E.1 gives the well known Paschen s euation for the breakdown potential. V b Bpd A ln( pd) + ln ln γ i (4) In large gaps, secondary emission by ion impact is the dominating phenomenon. However, at smaller gaps and large electric fields, another phenomenon called field emission becomes relevant. The theory behind this phenomenon is described in the classic paper by Fowler and Nordheim [13] based on the principle of uantum tunneling. With a sufficiently high electric field, the potential barrier at the surface of the cathode is thinned, thus facilitating the tunneling of electrons through the barrier. Starting with the Schroedinger s euation, Fowler and Nordheim were able to derive an expression for the electron emission current density, given by, 1.

3 A j FN ( β E) B v f FNφ 3 / ( ) exp φ βe (5) where A FN (~ 1.54 x 1-6 A ev V - ) and B FN (~ 6.8 x 1 9 ev -3/ Vm -1 ) are constants derived in the theory, ϕ is the work function (~4 ev) and v(f).6 and β is a field enhancement factor due to surface irregularities (~ 5-1 for most metallic surfaces). E is the applied electric field at the cathode surface. Boyle and Kisliuk (1959): In a simplified form, the field emission euation of Fowler and Nordheim (E. 5) can be written as, D E j Ce (6) where j is the current density of electron emission from the cathode due to field emission, E is the electric field and C and D are simplified constants. Starting with this form, Boyle and Kisliuk [6] made a series of assumptions to arrive at a final form of the electron emission coefficient (γ ) for ion-enhanced field emission. To consider the additional enhancement of the electric field due to the presence of a positive ion near the cathode surface, the total electric field was assumed to be a superposition of the applied field (E A ) and the field due to the ions (E + ), i.e., E E A + E +. Further, to simplify the formulation using Taylor expansion, it was assumed that E + << E A, and hence only the first order terms in E + were retained. As shown by Kisliuk in 1959 [1] (and shown in Fig.1), this assumption is not true as E + is comparable to E A. Fig. 1 plots the effect of the potential energy as a function of the distance from the cathode surface, with and without the presence of a single positive ion at a distance of 1 A o from the surface. The figure also includes a linear approximation to the ionenhanced field. As can be easily observed, the presence of the ion is a major contribution to the electric field and the linear approximation shows that the magnitude of the electric field is nearly doubled by the presence of the ion. Hence the assumption that E + << E A is incorrect. Boyle and Kisliuk s theory also makes an empirical assumption for the field enhancement due to the approaching ions. This enhancement is assumed to be a linear variation of the incoming ion current or E + Xj +. Also, the current density of electrons emitted from the cathode is assumed to be in power-law dependence with the incoming ion current density, i.e., j + Gj n E. n is a constant that is related to the other parameters at breakdown. The final form for the electron emission coefficient derived on the basis of the above assumptions is given by Potential Energy Ф(x) (V) Distance from cathode surface (A ) with the positive ion (E. 9) w/o the positive ion with ion - linearized (E. 11) Fig. 1 Field enhancement due to the presence of a single positive ion at a distance of 1 A from the cathode surface D EA γ ' Ke (7) The unknown parameter, K, is a result of the above mentioned assumptions. While the constant G is eliminated in the derivation, the field enhancement factor X and the constant n are retained in the final form of K. Despite all the above mentioned inaccuracies and limitations, Radmilovich- Radjenovic and Radjenovic showed E. 7 to be a good approximation to the emission coefficient at small gaps. Substituting E. 7 in the Townsend s criterion (E. 1), they derived a transcendental euation relating the breakdown potential to the gas pressure and gap distance. However, they failed to take into account the ionimpact emission coefficient (γ i ) which resulted in their expression being valid only for very small gaps (< 5 µm). Go and Pohlman [11] consider both forms of electron emission and superposing the two coefficients, formulate the total electron emission coefficient as γ γ i + γ ' (8) Substituting this into Townsends s euation gives a transcendental euation for the breakdown potential that is valid across the whole range of gap distances. However, the parameter K carries over into all the later extensions of Boyle and Kisliuk s theory. Curve fitting to experimental and computational data from literature, Go and Pohlman estimated K to vary between 1 6 and 1 8. PROPOSED MODE OF ION-ENHANCED FIED EMISSION The present model is inspired by the work of Kisliuk in 1959 [1] and other similar works on ionenhanced field emission in arcs. This section describes the model in detail, first abstractly and later 1.3

4 introducing the mathematics to make the arguments more rigorous. As mentioned above, when a sufficiently high electric field is applied between two electrodes, the thinning of the potential barrier allows the electrons to tunnel through it and be emitted from the cathode surface. The electrons enter the gas and undergo collisions with the gas molecules creating positive ions. These ions travel back to the cathode under the influence of the applied field. Once the distance between the incoming ion and the cathode surface reduces to a few nanometers, the electric field due to the ion near the surface begins to attain prominence. As the ion moves in closer to the surface, this enhancement grows higher causing a steady increase in the number of electrons ejected by field emission. This influence of the ion on field emission continues till the ion approaches a critical distance ( c ) at which it gets neutralized by an electron from the cathode. Thus during the life time of an ion between its creation and neutralization, it gives rise to a steady increasing yield (γ ) in electron emission hence the term ion-enhanced field emission. The neutralization process of the ion near the cathode surface may cause a second electron to be emitted by the Auger processes. These electrons are accounted for in the traditional secondary electron emission coefficient (γ i ). If a single ion with a unit positive charge is present at a distance from the surface of the cathode, at a distance x from the surface, the potential energy Ф + (x) due to the ion is given by Φ ( x) 4πε x + x (9) where is a unit positive charge, ε is the permittivity of free space. inearizing E. 9 for small x (i.e., x << ), gives ( x) πε x Φ + (1) Hence the total potential energy due to the applied field and the positive ion is given by x ( x) ( β EA) x + πε Φ (11) from which the total electric field at a distance x from the surface, E(x) is obtained as E( x) ( β E A ) + πε (1) The linearization of the potential energy is necessary since Fowler-Nordheim s derivation for the field emission electron current density reuires a constant electric field close to the surface. E. 11 is plotted in Fig. 1 for 1 A. When the effect of the ion s electric field on the cathode surface is extended to three dimensions with the problem being axisymmetric about the line of approach of the ion, the electric field near the cathode surface varies as Cathode Surface x r Positive Ion ( +r ) Fig. Definition of variables used in the derivation E( r) ( E A ) + πε β (13) ( + r ) 3 / where r is the distance along the surface from the line of approach of the ion as shown in Fig.. It should be noted that the field enhancement factor (β) has been taken to act only on the applied field. This is justified since the ion, being too close to the cathode surface, does not see the effect of the surface irregularities. As has been mentioned above, the ion while moving towards the cathode causes a steady increase in the field enhancement. This is clear from E. 13, wherein E increases as decreases. The distance at which an ion is present at any particular instant can be given as ( ) (14) t vt where is the distance at which the ion is created and v is the velocity of ion. But the effect of the field enhancement is negligible for large values of and hence the actual value of is not significant. It was observed that for > 1 nm, the electric field enhancement was negligible compared to the applied field and hence a constant value of 1 nm was chosen in the calculations. The velocity of approach of the ion can be related to the applied field (E A ) by the ion mobility through air (b) as v be A (15) Substituting E. 14 and E. 15 into E. 13 and then substituting for the total electric field in E. 5 gives an integral in time for the total electron current density due to the effect of a single positive ion. Also, since the electric field of the ion is now acting on the entire electrode, we can integrate over the area of the electrode to get the total number of electrons emitted by a single ion. This is in effect the definition of γ and is given by, 1 γ ' R T (πrdr) AFN E dt φ B exp FN 3/ φ v( f ) E (16) 1.4

5 In the above euation, E is a function of r and t since is a function of t. Since the effect of the ion s electric field decays as r increases, a numerical limit of R 5 nm was used. As has been mentioned before, the ion has a critical closest distance of approach ( c ) which has roughly been approximated as 1 A for the calculations. Hence the limits for the time integral is given by T c (17) The above integral in E. 16 has to be numerically integrated because of its complicated dependence on the variables. By substituting E. 16 into E. 8 and then the total emission coefficient into Townsend s ionization criterion (E. 1) and noting that E A V/d where V is the applied potential, Townsend s criterion can be numerically solved to obtain the breakdown potential (V b ). RESUTS AND DISCUSSION All numerical calculations were performed using Mathematica. Fig. 3 plots the breakdown voltage obtained from the above model as a function of the gap distance. Also included in the figure is the traditional Paschen s curve from E. 4 and the curve obtained from using the euation from Go and Pohlman [11] with a value of K 1 7. As can be seen, the present model behaves exactly like the curve obtained from the Go and Pohlman model and matches Paschen s curve at large gaps. This is in keeping with the physical mechanism of the entire process since the contribution of field emission progressively reduces as the gap increases. For all curves, the values for A and B were taken to be those of air [3] and β was taken as 5 and ϕ was taken as 4 ev. Breakdown Voltage, V b (V) Gap Distance (µm) v Rakshit (present model) Go and Pohlman K 1e7 Paschen's Curve Fig. 3 Plot of the breakdown potential (V b) as a function of the gap distance (d) As evident from Fig. 3, the present model also displays the three characteristic regimes in the curve the small gap rise, the plateau around 7 µm and the large gap Paschen s euation regime. Fig. 4 plots the ratio γ /γ i for various gap distances. As observed from the plot, the contribution of ion-enhanced field emission to the total emission coefficient is considerable at small gaps and slowly decreases at large gaps where only the secondary emission coefficient (γ i ) is significant. It should also be observed that the contribution of field emission has a sudden dip around 7 µm, dropping by about -3 orders of magnitude within a span of 3 µm. Because of the complexity of E. 16, the reason for this sudden drop is not obvious, but the drop explains the plateau region and the transition to the large gap Paschen s law regime. iterature review of experimental measurements of the breakdown potential (collected in [11]) show a broad variation, especially for small gaps (< 5 µm). As such, it becomes essential that the present model is capable of emulating such variations by the appropriate choice of electrode surface material and gas constants. Fig. 5 plots the breakdown potentials using the present model for different values of the field enhancement factor (β) and for two choices of work function (ϕ). As observed from the figure, the present model is able to accommodate a wide range for the breakdown potentials and shows sufficient variation in the position of the peak and plateau. γ'/γ i 1.E+ 1.E+1 1.E+ 1.E-1 1.E- 1.E-3 1.E-4 1.E Gap Distance (µm) Fig. 4 Ratio of γ /γ i as a function of the gap distance (d) CONCUSION The model presented above successfully eliminates the need for the empirical fitting parameter (K) that recurs in all the models based on the original theory by Boyle and Kisliuk. The present model is dependant only on the gas constants and electrode material and surface characteristics. While factors like β are more or less empirical, they are known to within acceptable limits and sufficiently well studied. Another factor used in the model is the distance of closest approach ( c ). While the effect of this parameter has been found to be insignificant (the 1.5

6 change being less than 3% in the breakdown potential), further studies will try to relate this parameter to the surface emission characteristics via the Auger theory of secondary emission. Breakdown Voltage, V b (V) beta 1 phi 4.5eV beta 5 phi 4eV 5 beta 1 phi 4eV Gap Distance (µm) beta 5 phi 4.5eV beta 6 phi 4.5eV beta 8 phi 4.5eV Fig. 5 Plot of the breakdown potential (V b) as a function of the gap distance (d) for different values of β and ϕ 8. Torres, J.M., Dhariwal, R.S., 1999, Electric field breakdown at micrometer separations in air and vacuum, Microsystem Technologies, 6, Radmilović-Radjenović, M., ee, J.K., Iza, F., Park, G.Y., 5, Particle-in-cell simulation of gas breakdown in microgaps, Journal of Physics D: Applied Physics, 38, Radmilović-Radjenović, M., Radjenović, B., 8, An analytical relation describing the dramatic reduction of the breakdown voltage for the microgap devices, Europhysics etters, 83,. 11. Go, D.B., Pohlman, D.A., 1, A mathematical model of the modified Paschen s curve for breakdown in microscale gaps, Journal of Applied Physics in press. 1. Kisliuk, P., 1959, Electron emission at high fields due to positive ions, Journal of Applied Physics, 3, Fowler, R.H., Norheim,., 198, Electron emission in intense electric fields, Proceedings of the Royal Society of ondon, 119, REFERENCES 1. Becker, K.H., Schoenbach, K.H., Eden, J.G., 6, Microplasmas and applications, Journal of Physics D: Applied Physics, 39, R55-R7.. Go, D.B., Fisher, T.S., Garimella, S.V., Bahadur, V., 9, Planar microscale ionization devices in atmospheric air with diamond-based electrodes, Plasma Sources Science and Technology, 18, Raizer, Y.P., 1991, Gas Discharge Physics, Springer-Verlag, Berlin Heidelberg. 4. Young, D.R., 1951, Electric Breakdown in CO from ow Pressures to the iuid State, Journal of Applied Physics, 1,. 5. Germer,.H., Haworth, F.E., 1948, A ow Voltage Discharge between Very Close Electrodes, Physical Review etters, 73, Boyle, W.S., Kisliuk, P., 1955, Departure from Paschen s law of breakdown in gases, The Physical Review, 97, Slade, P.G., Taylor, E.D.,, Electrical breakdown in atmospheric air between closely spaced (. µm-4 µm) electrical contacts, IEEE Transactions on Components and Packaging Technologies, 5,

Field Emission-Driven Microdischarges

small scale transport research laboratory Field Emission-Driven Microdischarges Prof. David B. Go dgo@nd.edu http://www.nd.edu/~dgo Aerospace and Mechanical Engineering 03/16/2012 Microplasmas and Microdischarges