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1. Introduction

Electrical breakdown and discharges in air have been subjects of intense research due to the interest in electrical phenomena in the Earth’s atmosphere [1, 2] and because of the use of air plasmas in a wide variety of technological processes [3–7], including surface treatment, pollution control, material pro-cessing, as well as in various biomedical applications.

Despite the advance of these application !elds, the starting phase of the electrical discharge, when the gas changes from insulating to a conducting state, is still of fundamental interest, as it has been, from the beginnings of gas discharge research. For gas breakdown in direct-current (DC) !elds, in his pioneering work [8], Paschen has derived a similarity law according to which the breakdown voltage is a function of the gas pressure × electrode distance, pL, product. This law is based on (i) the assumption that electrons multiply exponen-tially on their way from the cathode to the anode according

to I = I0exp(α x), where α is the Townsend ionization coef!-cient and (ii) an analytical approximation for the dependence of α on the electric !eld to gas pressure ratio, E/p, given as α/p = Aexp[−B/(E/p)]. Here A and B are constants, having spe-ci!c values for different gases [9]. Despite the simpli!cations of this theory, the Paschen law performs surprisingly well for many gases and for a relatively wide range of conditions. We note that the proper variable to be used in the Paschen law is the reduced electric !eld, i.e. the electric !eld normalized by the gas density, E/n, and not the electric !eld normalized by the pressure, as often quoted. (The unit conventionally used for the reduced electric !eld is 1 Td = 10−21 V m2.) The use of E/p is justi!ed at !xed temperature only.

Concerning air, measurements and models of electrical breakdown have been reported by several authors [6, 10–12], for different electrode materials and electrode spacings. Due to the more recent interest in atmospheric-pressure plasmas, the breakdown phenomena have also been investigated at

Journal of Physics D: Applied Physics

Experimental and kinetic simulation studies of radio-frequency and direct-current breakdown in synthetic air

I Korolov, A Derzsi and Z Donkó

Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, Konkoly-Thege Miklós str. 29-33, H-1121 Budapest, Hungary

E-mail: donko.zoltan@wigner.mta.hu

Received 27 June 2014, revised 17 September 2014Accepted for publication 30 September 2014Published 5 November 2014

AbstractWe report a combined experimental and modeling study of the breakdown of synthetic air under the in%uence of static (dc) and time-varying electric !elds, up to radio-frequencies (RF). The simulations of the breakdown events are based on a simpli!ed model that makes the problem computationally tractable: only electrons are traced by Monte Carlo simulation. In the dc case the simulation is used for the determination of the effective secondary electron emission coef!cient, based on the measured dc Paschen curve. In the RF case we observe a transition between the modes (i) where reproduction of the charges takes place in the gas volume (at electron oscillation amplitudes smaller than the gap size) and (ii) where surface processes—secondary electron emission and electron re%ection—play an important role, as most of the oscillating electrons reach the electrode surfaces. The RF (f = 13.56 MHz) simulations, assuming a constant effective secondary electron yield, reproduce well the breakdown curve obtained in the experiment.

Keywords: breakdown, Monte Carlo simulation, synthetic air, radio-frequency

(Some !gures may appear in colour only in the online journal)

I Korolov et al

Experimental and kinetic simulation studies of radio-frequency and direct-current breakdown in synthetic air

Printed in the UK

475202

JPHYSD

© 2014 IOP Publishing Ltd

2014

47

J. Phys. D: Appl. Phys.

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0022-3727

10.1088/0022-3727/47/47/475202

Paper

47

Journal of Physics D: Applied Physics

IOP

0022-3727/14/475202+9$33.00

doi:10.1088/0022-3727/47/47/475202J. Phys. D: Appl. Phys. 47 (2014) 475202 (9pp)

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small electrode gaps [7, 13]. Besides the ‘classical’ plane-par-allel electrode con!gurations, other electrode geometries (e.g. needle-to-plane) have been considered in other works [14].

Understanding of the breakdown phenomena is now aided by the more recent advances in the physics of charged par-ticle swarms (see e.g. [15]). Monte Carlo simulations of the electrons' motion within the electrode gap revealed, e.g. the existence of an equilibration region near the cathode, which has an effect to reduce the ‘useful’ length of the gap for the multiplication of the charges [16], especially at low pL values, i.e. on the left-hand side of the Paschen curve. Another recent extension to the classical breakdown theory has emerged from studies of micro-discharges operating at high (up to atmos-pheric) pressures. Here, the Paschen law becomes violated due to processes that do not scale with pL, e.g. !eld emission from the cathode at small gaps and high voltages, affecting signi!cantly the left-hand side of the Paschen curve [17, 18].

A major revision of Townsend theory of gas breakdown (that originally considered electrons and ions only) has recently been accomplished by Phelps and Petrović [19], who determined the contributions of the different species to sec-ondary emission at the cathode, for a wide range of the reduced electric !eld, E/n. Their !ndings for Ar gas indicated that at low E/n (≲100 Td) UV and VUV photons play the dominant role in releasing electrons from the cathode. At increased E/n, electron emission due to ion impact becomes dominant, while the contribution of metastable atoms is in the order of ∼10%. At reduced !elds higher than several kTd, fast atoms (origi-nating from elastic ion-atom collisions) have the major con-tribution. Secondary electron emission due to the different species occurs via different mechanisms; in some of these only the potential energy matters, while in some others the kinetic energy of the species also plays a role [20]. We also note that impurities, or even the presence of the buffer gas itself on the electrode surface (at otherwise clean conditions) may signi!-cantly change the secondary yields [21]. Secondary electrons can also be produced by electrons [22]. The threshold for this processes is around ∼50 eV and electrons with a few 100 eV may liberate even multiple electrons from surfaces. This pro-cess may lead to an ef!cient multiplication of electrons at high electron energies. For the conditions covered here, however, this process is expected to be negligible.

The contributions of the different species mentioned above to the secondary electron emission can be determined from accu-rate calculations, but only in cases when the energy-dependent electron yields for each relevant process are known [19]. Considering breakdown, the dc Paschen curve can be calcu-lated in a straightforward way if the above (microscopic) data are available. Unfortunately, availability of data for the energy-dependent electron yields for different gas/metal combinations is very limited. In such cases a combined experimental and mod-eling study can provide the effective secondary electron yield, γ*, which is de!ned as the ratio of the electron current to ion current at the cathode and accounts implicitly for the contribu-tion of other species mentioned above. Although in this case the Paschen curve cannot directly be computed from elementary data, the above procedure still provides some information about the possible electron emission processes through the dependence

of γ* on the reduced electric !eld. In our work we follow this approach (as in [4]) as accurate secondary yields for all relevant species and mechanism in an air plasma are largely unknown.

Besides studies with dc !elds, breakdown of air under the effect of time-varying (AC) !elds has also been studied from the low (audio) frequencies, through the radio-frequency domain, up to microwave frequencies [5, 23]. Such experiments, as well as the related modeling calculations are more complicated com-pared to the dc scenario; the relation of the characteristic time-scales of the ac !eld and of the charged particle motion de!nes the proper modeling approach to be used. Particle based kinetic simulations are applicabe for a wide domain of conditions. Most of such studies have been carried out for noble gases (e.g. for argon [24, 25]) and we are not aware of any previous work using such an approach for the case of air.

Concerning the domain of radio-frequencies, electrons follow the time-dependent electric !eld, while ions do not. For conditions when the electron oscillation amplitude in the RF !eld is smaller than the electrode gap (that happens at high frequencies and high pressures), gas phase charge reproduc-tion is dominant and the effects of surfaces are negligible. The key to an accurate modeling of breakdown under these condi-tions is a precise description of gas-phase processes. When the above-mentioned length scales become comparable (at low pressure and/or lower frequency), surface processes start playing a role, too. The dc case represents an ‘extreme’ limit when surface processes, such as secondary electron emission and electron re%ection from the surfaces, are indispensable. Due to the uncertainties related to surface processes we expect that the domain of gas phase charge reproduction is easier to model, as the electron impact cross sections for the constitu-ents of air are quite well known. For modeling the domain where surface processes are important, we shall use the results obtained for γ* in studies of the dc case.

In section 2 we describe the experimental setup, the methods of the measurements and present the experimental data and their comparison with results obtained in earlier studies. Section 3 describes the model of the dc and RF breakdown and compares the results of the calculations with the experimental data. A short summary of the work is given in section 4.

2. Experimental

In this section  the experimental setup is described !rst, fol-lowed by the presentation of the methods of measurements and the results of our experiments.

2.1. Experimental setup

The scheme of the experimental arrangement is displayed in !gure 1. Its base is a general purpose vacuum system equipped with %ow controllers, a cryogenic trap, a stainless steel vacuum chamber, valves to regulate the pressure and a tur-bomolecular + rotary pump system to establish vacuum. The breakdown cell is geometrically symmetric, made of glass and contains two stainless steel disk electrodes with D = 7.5 cm diameter, facing each other at a distance of L = 1.0 cm. While

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normally the discharge cells studied are placed inside the vacuum chamber, for the present measurements the cell is sit-uated outside the chamber, in order to minimize stray capaci-tances that could introduce an asymmetry and in%uence the breakdown processes at radio-frequency excitation.

The distance of the cell from metal parts of the vacuum system is more than 30  cm. The cell is connected to the vacuum system with 6 mm diameter glass tubes. Before per-forming the experiments, the cell is pumped down by a turbo-molecular pump to ⩽10−6 Torr. We use 5.0 purity synthetic air (79.5% of N2, 20.5% of O2, ⩽5 ppm of H2O, ⩽1 ppm CnHm), at a slow (3–8 sccm) %ow, regulated by a %ow controller. The gas is fed through a cryogenic trap (!lled with liquid nitrogen) to remove any remaining impurities. The gas pressure is meas-ured by a Pfeiffer CMR 362 capacitive gauge connected with a glass tube to the cell—the pressure measured this way is expected to be insensitive on the %ow.

The only drawback of using a glass-metal construction for the cell is that some of the parts had to be !xed with low vapor pressure epoxy (Torr Seal), disabling us to bake of the system, which would have otherwise been our standard procedure for any cell situated inside the vacuum chamber. This way, impu-rities (possibly dominantly water vapor) adhering to the glass surfaces may represent an issue. The electrode surfaces are cleaned by running moderate current (∼5 mA) dc discharges with both polarities for a time in the order of 10–20 min, before starting the breakdown measurements. The cleanliness of the cell is monitored by acquiring the optical emission spectrum, using an AvaSpec-3648 type spectrometer (having a relatively low resolution of ∼0.2 nm, but covering a wide spectral range of 250–800 nm). In the experiments we have not observed notice-able traces of water vapor (OH molecular bands) in the cell.

The experiments are controlled with a computer using LabView software. For the dc measurements the high voltage is established by a PS325 (Stanford Research Systems) power supply, interfaced with the computer via GPIB. For the radio-frequency measurements in a range of 0.1–13.56  MHz we have built a high-voltage oscillator, following a design devel-oped originally for ion guides and ion traps [26]. The oscil-lator operates with a pair of 6146B electron tubes and provides a symmetrical output, which is connected to the breakdown cell with short,  ∼20  cm, wires. The RF voltage amplitude can be set by the dc supply voltage of the oscillator: two TTi PLH-250P power supplies are connected in series (in order to achieve higher voltage amplitudes) and are controlled from the computer via an USB port. For the frequency range of 50 Hz–50 kHz we use ZXS 2000 power ampli!er connected to the cell through a transformer. The voltage amplitude and the fre-quency are controlled via the PC, using a function generator (Agilent 33220A). The actual voltage on the cell is measured by two voltage divider probes (Agilent 10076B), connected to two inputs of a digitizing oscilloscope (Picoscope 6403B), which sends the measured waveforms to the computer. The probes have been calibrated for the frequency range used in the measurements using a calibrated function generator. Depending on the frequency the uncertainty is in a range of 2% (<5 kHz)–7% (>8 MHz).

Breakdown events are ‘recognized’ by several methods: (a) the detection of the light of the plasma using a broad band (200–1000 nm), high sensitivity, low noise photodiode (58–262 Edmund optics). The signal from the photodiode is ampli!ed and sent to a discriminator, which is connected to the com-puter. (b) A change of frequency of the high voltage oscillator (or sudden drop of the voltage amplitude) due a change of the

Figure 1. Experimental setup for the breakdown voltage measurements.

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impedance of the cell. (c) Observing an increase of the power consumption of the RF oscillator (indicated by the output current of the dc power supply connected to the generator, recorded with a Metex 6000M multimeter). For methods (a) and (b), as it has been mentioned above, the waveform (typi-cally 3 × 104 samples covering 100 RF cycles) from the oscil-loscope, the information about current and the voltage of the power supply are sent to the computer, where they are analyzed using LabView software. Preceding the measurements, for the given frequency, we have scanned over possible voltage ampli-tudes, VRF, when the discharge cell is under vacuum conditions. During the scan, different parameters such as voltage wave-form, current and voltage of the power supply versus VRF have been recorded. Typically, the frequency stability or deviation from the set frequency for different voltage amplitudes is not more than 0.2%. During the breakdown measurements a devia-tion of the parameters from those measured in vacuum can be detected with less then 1% precision. The measured waveform is also !tted with a sine function and the deviation from the sine like curve during the breakdown (BD) event, especially for low frequency (<5 kHz) measurements, is detected.

The control program monitors all the three conditions and recognizes the breakdown event when any of these conditions is met. At frequencies above 4 MHz all the conditions were found to be ful!lled simultaneously within the time resolu-tion of our control program. At lower frequencies the light emission from the system was too weak to be detected, or the change of the current of the power supply was too low to be detectable, however, even for such conditions two of the three methods have indicated breakdown simultaneously.

Typically, each point on the BD curve is averaged from 10 points, which are scattered within 0–2%. Day-to-day repro-ducibility of the whole curve (tested for 13.56 MHz within 3 month period) is not more than 2%. The noise level of the recorded VRF is less than 1%. The estimated total error of the breakdown voltage measurements (including the voltage probes' uncertainty) is around 10% for the RF case and 5% for the dc case. The total harmonic content of the RF waveforms generated is less than 0.5%.

2.2. Measurement methods

The most traditional method to measure the breakdown voltage proceeds via a slow increase of the voltage applied to the electrodes, at a !xed gas pressure. In the following we will call this technique as the ‘voltage ramp’ method. As for the initiation of breakdown an external source of charged par-ticles is needed (cosmic radiation, UV light, etc) this method suffers from a statistical error, as breakdown does not nec-essarily occur when the theoretical breakdown voltage is already applied. In the limit of a low voltage ramp, however, this uncertainty is signi!cantly reduced. In our measurements we use voltage ramps in the 0.5–1.5 V s−1 range, with which breakdown voltage values within a few Volts are obtained in repetitive measurements at otherwise same conditions.

On the left hand side of the breakdown curve the voltage ramp method becomes increasingly inaccurate as the slope of the curve increases. Here, an alternative technique, the ‘gas

!lling’ method can be used, when a !xed voltage is applied to the cell and the gas pressure is slowly increased. Just as in the voltage ramp method, the slow increase of the pres-sure is crucial, the applicable rate can be found by requiring that repetitive measurements show small scattering of the data. This method also makes it possible to determine break-down voltages when the Paschen curve becomes multi-valued (breakdown occurs at more than one value of voltage at !xed pressure).

A further alternative method [27] identi!es the breakdown voltage as the sustaining voltage of a Townsend discharge in the limit of zero current. This method is applicable to the dc case only, it requires establishing a steady-state low-current Townsend discharge with a high voltage source and a high resistance in the electrical circuit. By decreasing the source voltage gradually, the voltage–current characteristics of the cell can be recorded. We also adopt this technique in our work and measure steady-state V  −  I characteristics at 0 ⩽ I ⩽ 1 − 2 µA.

With the voltage ramp method we measure 10 breakdown events, with the gas !lling method 5 breakdown events and present the average values of the measured quantities. We note that our cell allows to shine UV light at the electrodes to make the experiments even more reproducible, however, as we were able to establish stable conditions and reproducible results even without this external illumination, in the !nal experi-ments we have not used the external UV source.

2.3. Experimental data

The experimental data obtained for dc conditions are shown in !gure 2(a), together with data obtained in earlier studies. The measurements in synthetic air of Klas et al [4] have been executed with Cu electrodes and electrode gaps between 0.002 mm and 0.25 mm. Carr [11], Druyvesteyn and Penning [12] collected data for different electrodes in air, while Lisovskiy and Yakovin used duraluminum electrodes [6]. Our data obtained with the three different methods outlined above (voltage ramp ‘VR’, gas !lling ‘GF’ and low current limit ‘LCL’, as indicated in the !gure) show a high level of consistency. The data are also in a good agreement with those presented in [4], differences—that can be explained by the dif-ferent electrode materials—can be seen only in the left hand side of the Paschen curve. The measurements give a location for the Paschen minimum, pL ≈ 0.5 Torr cm and a minimum breakdown voltage, VBR,min  ≈  410  V. In the other previous works about 50 V lower breakdown voltage has been found, at a slightly higher pL value [6, 11, 12].

Figure 2(b) shows the voltage–current characteristics of the cell, for a series of gas pressures, recorded in the measure-ments when the breakdown voltage of the cell is identi!ed as the limit of the discharge voltage in the I → 0 limit (low current limit method, see above). The characteristics appear to be completely %at, just as one expects for a low current Townsend discharge.

Next, we illustrate the breakdown characteristics obtained at different frequencies, f, in !gure 3. In the experiments the two electrodes are driven by the generator with potentials

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U1 = U0 sin (2π f t) and U2 = − U0 sin (2π f t), respectively and the breakdown voltage is de!ned here as the peak value of the difference of these two potentials, VBR = 2 U0.

At f  =  13.56  MHz, the breakdown curve exhibits a par-ticular shape due to the onset of surface processes at lower gas pressures, when the amplitude of the electrons' oscilla-tions becomes comparable to the gap length [28]. For this frequency most of the data points have been recorded using the RF voltage ramp method, however, the nearly vertical part of the characteristic could only be scanned by using the gas !lling approach. The shape of the breakdown curve for the f = 13.56 MHz case is very similar to that obtained by Lisovskiy and Yegorenkov [5] in air, although this work resulted in about 30% lower minimum breakdown voltage. As at the minimum breakdown voltage the charge reproduction is dominated by gas phase reactions, the breakdown voltage value is not expected to depend on the surface parameters, as it will be shown later on. Considering this fact, this relatively large deviation is not clearly understood, it is perhaps caused by constituents of air that are not present in synthetic air.

At f = 8 MHz, our data show a gap between 7 and 12 Torr, due to the maximum output voltage limitation of the RF gener-ator. At the lower frequencies studied, the ‘vertical transition’ between the regions of gas-phase-dominated and surface-dominated charge reproduction disappears. In the limit of low frequencies the breakdown characteristic converges towards the dc Paschen curve. At f ⩽ 1 kHz the discharge is ignited in each period and at each electrode independently of the pre-vious pulses and the voltage ramp is slow enough to ignite the dc discharge.

3. Simulations

3.1. Dc case

As already mentioned above, the lack of knowledge about the (energy-dependent) secondary electron yields of the

important species, which contribute to the charged particle reproduction at the cathode surface, makes it impossible to calculate an accurate dc Paschen curve for air from ‘elemen-tary’ data. Therefore, for the dc case we adopt the concept of using an effective secondary electron yield and calculate its value, as a function of E/p. In this approach only electrons have to be traced in the calculations. We accomplish this by Monte Carlo simulation [29], which provides a fully kinetic description under the conditions of nonlocal transport [30]. The electron trajectories are obtained by integrating their equations of motion (with a time step ∆t) in the presence of the applied !eld, =  m t q x tr E( ) ( , )

... The occurrence of col-

lisions is checked at every time step, for each electron. The probability of a collision in the cold gas approximation is: P (∆t) = 1 − exp[−n σT(v) v ∆t], where σT(v) is the total elec-tron–atom collision cross section and v is the velocity of the electron. The types of collisions and the scattering processes are treated in a stochastic manner [31].

Figure 2. (a) Breakdown voltages in dc mode measured by the three different methods (voltage ramp ‘VR’, gas !lling ‘GF’ and low current limit ‘LCL’), as well as Paschen curves obtained in previous studies [4, 6, 11, 12]. (b) Steady-state voltage–current characteristics in the Townsend regime, for various pressures.

Figure 3. Breakdown voltages obtained in the experiments at different frequencies, including the limiting, dc, case (voltage ramp ‘VR’, gas !lling ‘GF’ and low current limit ‘LCL’). The dashed line shows the results of Lisovskiy and Yegorenkov presented in !gure 2 of [5].

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The simulation is three dimensional in velocity space but is only one dimensional in real space, i.e. the !nite radial exten-sion of the cell is not taken into account (justi!ed by the rela-tively large diameter-to-gap ratio, D/L = 7.5). Although a full three-dimensional spatial description would not be dif!cult with Monte Carlo simulation, uncertainties would arise due to the incomplete knowledge of the interaction of the plasma species with the glass cylinder that surrounds the electrodes in the experiment.

In the simulation we trace electrons emitted from the cathode along their path to the anode. The number of the initial electrons, 104, was chosen as a result of the trade-off between minimizing the noise level in the simulation (to achieve better accuracy and reproducibility) and the code execution time. We consider an extensive set of e−-N2 and e−-O2 collisions, adopting the cross sections  used in [2]. Following the addi-tional electrons created in ionizing collisions, we simulate a high number of complete electron avalanches (N0), to reach a good level of statistics. We account for the re%ection of elec-trons from the electrode surfaces, as this process may become important for certain conditions: (i) electrons emitted from an electrode may be backscattered to the surface after elastic col-lisions with the gas atoms. Due to this effect some of the elec-trons will be re-absorbed at the surface thereby reducing the apparent secondary yield. The ‘escape factor’ of the electrons is especially reduced at low E/n, as shown in [19] (to about 10% at E/n ∼ 10 Td in argon). (ii) Re%ection of electrons is also important at conditions when they reach the surface with an appreciable energy; in this case re%ected electrons may pro-duce additional ionization [32]. The probability of re%ection depends in a complicated manner on the material and surface properties, on the energy of the electron, as well as on the angle of incidence. Precise data, especially such that cover a wide electron energy range, are dif!cult to !nd, therefore we apply a constant value for the elastic backscattering (re%ection) prob-ability of electrons from the electrode surfaces: ρ = 0.2 [33].

In agreement with the concept of using an effective sec-ondary electron yield (γ*) we assume that only ions con-tribute to electron emission and we also assume that all the ions created in the gas phase reach the cathode (due to their directed motion under the effect of the electric !eld). Using this approach, for any set of conditions we run the simulation with the measured value of breakdown voltage and determine the secondary yield from

γ =+ ∗N N ,0 (1)

where N+ is the total number of ions created in the simulation of electron avalanches initiated by the emission of N0 electrons from the cathode. In other words, at any !xed pressure and at the experimentally measured breakdown voltage we run a high number of electron avalanches and determine the number of ions reaching the cathode (by counting the number of ionizia-tion events and assuming that all ions reach the cathode under in%uence of the directed electric !eld, justi!ed by the dimen-sion of the cell). The ratio between the number of emitted elec-trons and incoming ions de!nes the effective secondary yield. Breakdown, in this sense, is de!ned as the state when a precise reproduction of charged particles is ful!lled.

The secondary yield values, γ*(E/p), obtained this way are shown in !gure 4. In the limit of low reduced electric !eld γ*(E/p) is in the range of ≅0.002 and increases quite signi!-cantly, by about two orders of magnitude towards the high E/p values. The present data agree reasonably well with those obtained by Klas et al [4], who have used Cu electrodes with signi!cantly smaller gaps, L ⩽ 0.25 mm.

3.2. Simulation of the RF case

The simulation of the RF case differs in one important aspect from the simulation of the dc case. While in the dc case we only counted the number of ions created in a given number of electron avalanches (to !nd the balance of charged particle loss and creation), in the RF simulations we follow the time dependent evolution of the electron density. The model is, however, still simpli!ed to a great extent, as we still trace a single type of particles: electrons. We note that this approach reduces very signi!cantly the computational time, as tracing of ions, e.g., would be extremely inef!cient as they move in a diffusive way under the effect of the high-frequency !eld, of which the time average (in absence of space charges) is zero. We handle secondary electron emission processes in the fol-lowing way: we emit an electron from either of the electrodes (chosen randomly) following the ‘birth’ of an ion, with a prob-ability γ fi at a random time within the next RF cycle. Here, fi is the fraction of ions that reaches either of the electrodes with diffusive motion, which is approximated by considering only the lowest order diffusion mode in the cylindrical geometry of the cell:

=+

π

π ( )( )

( )f ,i

L

L R

2

2 2.405 2 (2)

where R is the radius of the electrodes.The RF simulation at a !xed set of conditions starts with

seeding N0 electrons in the center of the electrode gap. These

Figure 4. Effective electron yield, γ*, as a function of E/p. The dashed line represents averages for the different gap sizes studied by Klas et al, as shown in !gure 2 of [4].

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electrons are traced as they move under the in%uence of the RF !eld, until a de!ned time limit of the simulation (tmax) is reached, or they reach either of the electrodes.

For f = 13.56 MHz the simulation time is set to 500 RF cycles, tmax = 500 T (where T is the length of the RF period). The number of electrons, Ne(t), is continuously monitored during the simulation and a straight line is !t to Ne(t) during the second half of the run, i.e. for cycles 250–500. This !t suppresses the periodic variation of the number of elec-trons within the RF cycles and gives information about the change of their number over a suf!ciently long time scale. The breakdown event is associated with a slope near to zero, that expresses a periodic steady-state of a very low density plasma. Similarly to the dc case, the voltage is increased in small steps in consecutive simulation runs to !nd the break-down balance at !xed pressure (voltage ramp method), or the pressure is increased in small steps at !xed voltage (gas-!lling method).

In !gure 5(a) we illustrate the evolution of the number of electrons at a series of voltage amplitudes, for f = 13.56 MHz and p  =  10  Torr. The breakdown balance is found at VBR = 483 V. A few Volts difference in the amplitude already results in a signi!cant positive or negative slope, thus we con-clude that the accuracy of the determination of VBR is within a few Volts. Panels (b)–(d) of the same !gure  illustrate the spatio-temporal evolution of the electron density at the break-down voltage (b), as well as at VBR − 5 V (c) and VBR +5 V (d).

Here, the distributions are plotted up to t = 200 T. The nearly steady electron density in !gure 5(b), a decaying density in !gure 5(c) and an increasing density in 5(d) is easy to observe.

It is also obvious from !gure 5(b) that the electron density is rather low in the vicinity of the electrodes, their oscilla-tions occur near the mid plane of the gap. In order to resolve these oscillations, !gure 6 shows only !ve cycles for selected points on the breakdown curve, covering the range of pres-sures between p = 0.6 Torr and 20 Torr. At the highest pres-sure (see !gure 6(a)) the amplitude of electron oscillations is small, compared to the gap size. When the pressure is low-ered to 10 Torr (panel (b)), the oscillation amplitude slightly increases. Panel (c) of the same !gure  corresponds to the minimum breakdown voltage. At the corresponding pressure of 2.56 Torr we observe well-developed electron avalanches between the two electrodes, with signi!cant electron multi-plication during each half of the RF cycle. At further lowered pressure the qualitative picture of the self-sustainment remains the same, but the electron density gets more concentrated near the electrodes, as a consequence of the very high rate of multiplication in the avalanches bridging the electrodes. The majority of the electrons created in these avalanches reaches the electrodes and gets absorbed. However, a small fraction (originating from surface re%ection) stays at the vicinity of the given electrode and upon the change of direction of the electric !eld these electrons give rise to the next avalanche directed towards the opposite electrode.

Figure 5. (a) The number of electrons in the simulation as a function of time, Ne(t), at different voltages. p = 10 Torr. The breakdown event is identi!ed as a nearly constant electron number in the !tting interval, which is between 250 and 500 cycles in the case of f = 13.56 MHz. (b)–(d) Spatio-temporal evolution of the electron density (in arbitrary units) during the !rst 200 RF cycles for (b) 483 V, corresponding to the breakdown event, (c) for 5 V lower applied voltage and (d) for 5 V higher voltage.

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The results of the simulations, adopting different sec-ondary yield (γ*) and surface electron re%ection values (ρ), compared with the experimental data are shown in !gure 7 for

f = 13.56 MHz. The computed breakdown voltages, in agree-ment with our observations on the electrons' motion, discussed above, are insensitive to γ* and ρ in the p > 2.5 Torr pressure range, as here the charged particle density is sustained by volume reactions. At lower pressures, the surface reproduc-tion of electrons plays a role too, and the breakdown curve gets sensitive on parameters that characterize the surface processes. The best overall agreement between the computed curve and the experimental data is reached at γ* = 0.002. This, quite low, value of the secondary emission coef!cient matches the low E/p limit of γ* derived in the dc case and shown in !gure 4. In the case when neither of the surface mechanisms are considered, the breakdown curve turns to be multi-valued and the p < 2.5 domain becomes inaccessible.

4. Summary

In this paper we have reported a combined experimental and kinetic simulation study of the breakdown of synthetic air under dc and RF excitation, the latter covering the f ⩽ 13.56 MHz frequency range. The experiments have been carried out at voltages up to 1200 V and pressures up to 20 Torr, using a cell with plane parallel stainless steel disk electrodes, separated by a gap of 1 cm.

Figure 6. Spatio-temporal distribution of the electron density (given in arbitrary units) during 5 RF cycles (from 35th RF cycle after starting the simulations), for f = 13.56 MHz, for points selected from the breakdown curve: (a) 20 Torr, 847 V, (b) 10 Torr, 483 V, (c) 2.56 Torr, 370 V and (d) 0.6 Torr, 829 V.

Figure 7. Comparison of the experimental breakdown voltage values with simulation results illustrating the effect of surface processes on the calculated breakdown characteristics.

J. Phys. D: Appl. Phys. 47 (2014) 475202

I Korolov et al

9

A simulation model has been developed that traces the electrons at the kinetic level and contains, on the other hand, serious simpli!cations to avoid the need for tracing the other species (ions, metastable atoms, photons) in the cell, that could have been realized only at extremely great computa-tional cost. Based on this model the effective electron yield, as a function of the reduced electric !eld, γ*(E/p), has been determined from the measured dc Paschen curve.

The RF breakdown curve has been well reproduced in the simulations using a constant effective electron emission coef!cient. The simulations made it possible to uncover the electrons' behavior across the different regimes of operation mode, closely connected to the oscillation amplitude of elec-trons in the RF !eld. The importance of surface re%ection of electrons has also been demonstrated.

Acknowledgments

We thank J-P Booth and V Lisovsky for directing our atten-tion to this topic, as well as Z Petrović and S Dudin for useful discussions on RF breakdown. Special thanks are due to Scott Anderson for advices on the construction of the high-voltage RF source. The contribution of P Hartmann to the construc-tion of the experimental set-up is gratefully acknowledged. This work has been !nancially supported by the Hungarian Scienti!c Research Fund, via Grant OTKA K105476.

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