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Mechanism behind self-sustained oscillations in direct current glow discharges anddusty plasmas

Sung Nae Cho∗Devices R&D Center, Samsung Advanced Institute of Technology,

Samsung Electronics Co., Ltd, Mt. 14-1 Nongseo-dong,Giheung-gu, Yongin-si, Gyeonggi-do 446-712, Republic of Korea.

(Dated: 2 April 2013)

An alternative explanation to the mechanism behind self-sustained oscillations of ions in direct current (DC)glow discharges is provided. Such description is distinguished from the one provided by the fluid models, whereoscillations are attributed to the positive feedback mechanism associated with photoionization of particles andphotoemission of electrons from the cathode. Here, oscillations arise as consequence of interaction betweenan ion and the surface charges induced by it at the bounding electrodes. Such mechanism provides an elegantexplanation to why self-sustained oscillations occur onlyin the negative resistance region of the voltage-currentcharacteristic curve in the DC glow discharges. Furthermore, this alternative description provides an elegantexplanation to the formation of plasma fireballs in the laboratory plasma. It has been found that oscillationfrequencies increase with ion’s surface charge density, but at the rate which is significantly slower than it doeswith the electric field. The presented mechanism also describes self-sustained oscillations of ions in dustyplasmas, which demonstrates that self-sustained oscillations in dusty plasmas and DC glow discharges involvecommon physical processes.

I. INTRODUCTION

When an ion is confined between the two plane-parallelelectrodes and is subject to static electric field (see Fig.1), it begins to self-oscillate without damping. Such phe-nomenon is referred to as self-sustained oscillations in di-rect current (DC) glow discharges; and, it has been knownfor almost a century.1–7 In literature, the phenomenon ofDC glow discharge is also referred to as the DC glowcorona. The mechanism behind such oscillations is still notfully understood.8,9,11–17 Over the years, various theoreti-cal models have been proposed in an attempt to explain thephenomenon.5,11,18–26Among the successful ones are thosebased on the fluid and equivalent circuit models.

The equivalent circuit models try to predict oscillations byrepresenting the system with an equivalent RLC circuit, whereR is a resistor, L is an inductor, and C is a capacitor.18–21Al-though this approach is quite useful in describing oscillationfrequencies as function of DC bias voltages, it says nothingabout the mechanism behind self-sustained oscillations.

The fluid models approach the problem from more fun-damental grounds of the electromagnetic theory. In this ap-proach, the Poisson equation is solved in combination withthe electron and the ion flux continuity equations, which con-stitute the so called positive feedback mechanism.22–26Asser-tion of such feedback mechanism is crucial in the fluid modelsbecause, without it, no oscillatory solutions can be obtained.Typical sources of the positive feedback mechanism includethe photoionization of particles and photoemission of elec-trons from the cathode. Physically, such feedback mechanismpromotes oscillations by periodically reversing the particle’scharge polarity. An ion oscillating near an electrode inducescurrent in the same electrode, where the waveform of suchcurrent is correlated to its velocity profile.23 Experimentally,it is this induced electrode current (or voltage) which getsmeasured.6,11 The hallmark of the fluid models is their poten-tial to reproduce experimental measurements to a reasonably

good accuracy. In principle, with an appropriate specificationof the positive feedback mechanism, the level of accuracy pre-sented by the fluid models can always be improved. The feed-back mechanism varies among different authors; and, this hasbeen a subject area of ongoing debate among different fluidmodel theorists.11,22–26

σ2σ1

2κ3κ

xe is out of page

VL

VT

VT

VL

ze

dz

Ep

Cathode

Anode

a

b

h

s

ye

Conductor core

Dielectric shell

Figure 1: (Color online) Illustration of core-shell structured ion con-fined between DC voltage biased plane-parallel conductors with avacuum gap ofh. Ep =−ez (VT −VL)/h is the parallel plate electricfield, where (ex,ey,ez) are versors along the Cartesian (x,y,z) axes,respectively.

Does the aforementioned positive feedback mechanism,which is asserted in the fluid models, represent the funda-mental mechanism behind the phenomenon of self-sustainedoscillations in the DC glow discharges? This is a subtle ques-tion because I have shown recently that an ion confined be-tween the DC voltage biased plane-parallel conductors goesthrough an undamped self-oscillatory motion.27 Such self-oscillatory motion requires that an ion is electrically polar-izable, but it does not necessarily involve or require the dis-cussed positive feedback mechanism which is asserted in thefluid models. The fact that an ion must be electrically po-

2

larized excludes electrons from consideration in the discus-sion of self-oscillations presented in this work. However,theproton, which is known to be electrically polarizable,28 is ex-pected to self-oscillate in the DC glow discharges accordingto this alternative theory.

Remarkably, the predictions of this alternative theory29

qualitatively agree with the predictions made by the fluidmodels.23,25 Both theories predict a saw-tooth shaped wave-form for the induced currents in the electrodes. Furthermore,sharp pulses of radiation output are predicted by both theoriesto accompany the abrupt rises in the induced electrode cur-rents. Such remarkable similarities in the predictions by thetwo very dissimilar theories suggest that the discussed positivefeedback mechanism, which is asserted in the fluid models,may not necessarily represent the fundamental mechanism be-hind the phenomenon of self-sustained oscillations in the DCglow discharges.

In this work, I shall present some new aspects of this al-ternative theory29 for further exploitation of the phenomenonof self-sustained ion oscillations in the DC glow discharges.These theoretical predictions are explicitly compared withvarious experimental results for the validation of the model.By direct application of the model to describe the charged-particle oscillations in dusty (or complex) plasmas experi-ments, I shall reveal that the phenomenon of self-sustainedoscillations in both dusty plasmas and DC glow dischargesinvolves a common physical mechanism in which self-oscillations are attributed to the interaction between an ion-ized particle and the surface charges induced by it at the sur-faces of the particle confining electrodes. Considering thatparticle oscillations in both dusty plasmas and DC glow dis-charges experiments involve ionized particles that are onlydiffer in their physical sizes and masses, such outcome is nottoo surprising. After all, apart from this, everything elseisnearly identical from the physics point of view in both exper-iments.

II. BRIEF OUTLINE OF THE THEORY

The self-sustained oscillations in DC glow discharges arenow discussed briefly in the framework of presented alterna-

tive theory using the configuration illustrated in Fig. 1. Theelectric potential between the plates is obtained by solvingthe Laplace equation with appropriate boundary conditions.27

With this electric potential, induced surface charges at the sur-faces of conductor plates are obtained by application of theGauss’s law. Such charge distributions act on an ion and gen-erate resultant force given byFT = F1+F2−ezmg, wheremis the mass of an ion,g = 9.8m/s2 is the gravitational con-stant, andF1 (F2) is the force between the ion and the surfacecharges induced by it at the surface of anode (cathode). Theexplicit forms ofF1 andF2 are given by27

F1 = ezπε0κ3ν

ν4s2 +Ep

[

γ(

b3− a3)

− b3

4s3 −1

]

,

F2 =−ezπε0κ3ν

ν4(h− s)2

−Ep

[

γ(

b3− a3)

− b3

4(h− s)3−1

]

,

where ε0 is the vacuum permittivity,Ep ≡∥

∥Ep∥

∥ is theparallel-plate electric field strength, the parameters is the dis-tance from the ion’s physical center to the anode; and, thetermsγ andν are defined as

γ =3κ3b3

(κ2+2κ3)b3+2(κ2−κ3)a3 , (1)

ν =2a(b− a)σ1

ε0κ2+

a2σ1+ b2σ2

ε0κ3, (2)

whereσ1 (σ2) is the surface charge density at the ion’s core(shell) of radiusr = a (r = b), and the dielectric constantsκ2andκ3 are depicted in Fig. 1. The resultant force on an ion is,hence, given by

FT (zd) = ezπε0κ3ν

4

ν(zd + b)2

−ν

(h− zd − b)2+Ep

[

γ(

b3− a3)

− b3

(zd + b)3+

γ(

b3− a3)

− b3

(h− zd − b)3−8

]

−ezmg, (3)

wheres = zd + b in F1 andF2 (see Fig. 1). The potential energy associated with this force is given by29

U (zd) =πε0κ3ν

4

νzd + b

+ν

h− zd − b−

4νh

+Ep

[

γ(

b3− a3)

− b3

2(zd + b)2−

γ(

b3− a3)

− b3

2(h− zd − b)2+8

(

zd + b−h2

)

]

+mg

(

zd + b−h2

)

; (4)

3

and, the equation of motion associated with this force, i.e., Eq. (3), can be expressed as

d2zd

dt2 =πε0κ3ν

4m

ν(zd + b)2

−ν

(h− zd − b)2+Ep

[

γ(

b3− a3)

− b3

(zd + b)3+

γ(

b3− a3)

− b3

(h− zd − b)3−8

]

− g, (5)

whereγ andν are defined in Eqs. (1) and (2), respectively.27,29

III. COMPARISON TO THE EXPERIMENTS

Gases under pressure in general, including noble gases,are composed of atomic clusters, a state of matter betweenmolecules and solids, due to Van der Waals interaction.30–32

In gaseous argon, each spherical atomic clusters of radiusr ≈ 1nm contains roughly 135 argon atoms.31 Modeling par-ticles in the gas as atomic clusters is important because gasesare delivered to the laboratories in pressurized containers, inwhich environment individual particles in gas exist in the formof atomic clusters. That said, I shall explicitly work with anargon atomic cluster of radiusr ≈ 1nm, which for brevityis simply referred to as “ion” hereafter. Such ion is not ex-pected to be core-shell structured like the one depicted inFig. 1. The shell portion of an ion can be eliminated bychoosingb = a, κ2 = ∞, and σ2 = 0C/m2. For a positiveion, its surface charge density is given byσ1 = Nqe/

(

4πa2)

,whereN is the number of electrons removed from the parti-cle andqe = 1.602×10−19C is the fundamental charge mag-nitude. Without loss of generality, and for the purpose ofclear illustration in this work, I shall chooseN = 250 anda = 1nm. This corresponds to the ion’s surface charge den-sity of σ1 ≈ 3.19C/m2. Assuming the mass of an argon atomis 6.63×10−26kg, a spherical atomic cluster composed of 135argon atoms has a total mass ofm ≈ 8.95×10−24kg, wherethe masses of missingN electrons have been neglected. Purelyfor convenience, I shall assume that the cathode is groundedand the space between the two conductor plates in Fig. 1 isa vacuum with a gap ofh = 100nm. The obtained parametervalues are summarized here for reference:

κ2 = ∞, κ3 = 1, h = 100nm, VL = 0V,σ1 ≈ 3.19C/m2, σ2 = 0C/m2,

m ≈ 8.95×10−24kg, b = a = 1nm.(6)

Illustrated in Fig. 2 is a plot of the potential energy well,where Eq. (4) has been plotted forVT = 100V using theparameters defined in Eq. (6). The widthlD of the poten-tial energy well decreases with an increase in the electric fieldstrengthEp. Such characteristic provides an elegant explana-tion to the experimental observations in which the oscillationfrequencies increase with the applied electric field strength.Fox seems to be the first to discuss on such characteristic us-ing a parabolic potential model.3

In the case of positive ions, the potential energy well getsformed in vicinity of the anode at the presence of appliedstatic electric field. On the other hand, for the negative ions,the same electric field results in formation of the potentialenergy well in vicinity of the cathode. Such characteristic

−5−4.5

−4−3.5

−3−2.5

−2−1.5

−1−0.5

0 10 20 30 40 50

U(

) [k

eV]

hp

E pz d

ez

lD

zd [nm]

Figure 2: Potential energy is plotted forVT = 100V using parametervalues defined in Eq. (6). Here,VT = 100V corresponds toEp =1GV/m.

provides an elegant explanation to the phenomenon of fire-ball formation in the laboratory plasma. When a positivelybiased electrode is immersed in plasma, a region of intenseglow appears in vicinity of the electrode. Such glow forma-tion is referred to as a plasma fireball.33 In the framework ofthis alternative theory27,29 elaborated in this work, the forma-tion of such plasma fireballs can be elegantly explained fromthe potential energy well which gets formed near the anodefor positive ions. Since the dynamics of a plasma involvescollective motions of its constituent atoms, a weakly ionizedplasma fireball, as a whole, self-oscillates near the electrodeas if it were a weakly ionized single superparticle. Through-out this work, I shall use the term “superparticle” to refer tosuch entity as plasma fireball whose dynamics can be repre-sented by an equivalent single-particle picture. In the equiva-lent single-particle picture, the self-oscillation dynamics of aplasma fireball is described by the equation of motion definedin Eq. (5) with appropriate effective mass and surface chargedensities prescribed. With that in mind, the plasma fireballef-fectively has a very large mass and carries a net charge whichis very weak. Consequently, induced current oscillations ofrelatively low frequencies and small amplitudes are expectedto be generated at the anode near a self-oscillating plasma fire-ball. Such prediction is consistent with the experimental ob-servation by Stenzelet al.33 Notice that this alternative theorycan be directly applied to describe the self-oscillations of ion-ized dust particles in dusty plasmas. It is remarkable that theprofile of the potential energy well illustrated in Fig. 2 quali-tatively agrees with the measurements by Tommeet al.34 andArnaset al.35 in dusty plasmas. Further discussion on this isprovided in the sections A thru C of the appendix.

The particle’s equation of motion described in Eq. (5) canbe solved via Runge-Kutta method using the parameter values

4

defined in Eq. (6) and the initial conditions given by

zd (0) = 0.5nm anddzd (0)

dt= 0

nms. (7)

The results forVT = 50V, 75V, and 100V are shown inFig. 3(a), where it shows the oscillation frequencies increas-ing with the electric field strength. The oscillation frequen-cies also increase with the ion’s surface charge density, butat the rate which is significantly slower than it does with theelectric field strength (see Fig. 3(b)). For instance, whenthe electric field strength is doubled fromEp = 0.5GV/mto Ep = 1GV/m, oscillation frequency nearly doubles fromνosc ≈ 104GHz toνosc ≈ 191GHz. However, doubling theion’s surface charge density, i.e.,σ1 = Nqe/

(

4πa2)

, fromN = 250 toN = 500 only slightly increases the oscillation fre-quency fromνosc≈ 191GHz toνosc≈ 207GHz. Such charac-teristics is consistent with the observations by Fox3 and Bošanet al.,36 where they have reported that oscillation frequenciesdo not seem to be very dependent on the type of ions used inthe discharge. It is well known that dissimilar atomic gaseshave different ionization tendencies. Based on this, it canbeinferred thatσ1, in general, for different atoms under identicalconditions are different.

Although a relatively high static electric field is being ap-plied to the ion in Fig. 3(a), its kinetic energies are smalldue to the fact that ion is going through frequent turnings in-side a tight potential energy well. The ion’s kinetic energydecreases with increased electric field strength for that matterand this is illustrated in Fig. 3(c). At the maximum speed of∼ 2.912km/s, the ion gains kinetic energy ofEK ≈ 237eV,which is small compared to the depth ofph ≈ 4.1keV for thepotential energy well but large compared to the average roomtemperature thermal energy ofET = 1.5kBT ≈ 39meV (here,T = 300K andkB = 8.62×10−5eV/K is the Boltzmann con-stant). Self-sustained oscillations in the DC glow dischargescan thus be expected to have high thermal stability.

In the experiments, self-sustained oscillations of ions intheDC glow discharges occur only in the negative resistance re-gion of the voltage-current characteristic curve.12,14,18,24Suchnegative resistance region in the voltage-current characteris-tic curve is characterized byVth1 ≤ VT ≤ Vth2, whereVth1andVth2 are some threshold voltages. Equation (5) providesan elegant explanation to such properties. For instance, Eq.(5) yields solutions that are unphysical forEp < Ep,th1 andEp > Ep,th2, whereEp,th1 and Ep,th2 are the threshold elec-tric field strengths. The physical solutions are only obtainedfor Ep,th1 ≤ Ep ≤ Ep,th2. It is difficult to pinpoint Ep,th1 andEp,th2 without the analytical solution of Eq. (5) at hand. Nev-ertheless, these can be roughly estimated on the grounds ofphysicality. To illustrate this, the oscillation frequency is firstplotted as a function of the electric field strength in Fig. 3(d)by numerically solving Eq. (5) using the parameter values andthe initial conditions from Eqs. (6) and (7), respectively.Thesolutions obtained forEp . 0.1GV/m andEp & 4.9GV/mare unphysical. For instance, in the case ofEp & 4.9GV/m,solutions show that the ion penetrates into the anode’s surfacewhereas forEp . 0.1GV/m, solutions yield peak to peak am-plitudes that are larger thanh/2. This latter case, although

mathematically allowed, is unphysical because, for a posi-tive ion, oscillations can only exist forzd < h/2− b fromthe argument based on the grounds of physicality.27 Hence,the physical oscillatory solutions exist only for 0.1GV/m .Ep . 4.9GV/m, where the upper and the lower bounds ofEpare only rough estimates based on the grounds of physicalityarguments. Further discussion on the properties of the phys-ical and the unphysical oscillatory solutions are providedinthe section D of the appendix.

Recently, Lotzeet al.37 reported on an indirect account ofoscillations involving a single H2 molecule in the DC volt-age biased conductors near an absolute zero temperature ofT = 5K. According to their findings, the H2 molecule in thejunction, which is the space between the atomically cleanCu(111) surface and the STM (scanning tunneling micro-scope) tip mounted on a cantilever, self-oscillates betweenthe two unknown positional states when a threshold DC biasvoltage is applied to the electrodes, i.e., the copper Cu(111)surface and the STM tip. The footprint of self-sustained os-cillations is the presence of the negative differential conduc-tance in their measurement. Guptaet al.38 showed that suchnegative differential conductance is due to the H2 moleculesin the junction. In the DC glow discharges, self-sustainedoscillations arise as a consequence of the negative differen-tial resistance.10,12,14Since the negative differential conduc-tance and the negative differential resistance are reciprocallyrelated, they represent an equivalent description of the samephysical processes. It is well known that the H2 moleculescan be ionized by a process of an electron impact.39 Whena threshold DC bias voltage is applied to the electrodes, theenergetic electrons begin to tunnel through the junction. Itis possible that the H2 molecules in the junction are ionizedin the process. If that is indeed the case, what Lotzeet al.reported may be an indirect account of self-sustained oscilla-tions involving a single or a few ions in the DC glow corona.29

Their observation can serve as a catalyst for the future exper-iments in the few-particle DC plasma experiments.

Arnas et al.40 reported that a dust particle made of hol-low glass microsphere of radiusr ≈ 32µm with mass densityof ρm ≈ 110kg/m3 and carrying a total electrical charge ofQ≈−4.3×105qe,whereqe =1.602×10−19C, self-oscillatesat a frequency ofνosc≈ 17Hz inside the plasma sheath. Theplasma sheath environment, in many respects, is similar tothe empty space region between the DC voltage biased plane-parallel conductors. Ionized particle oscillations in theplasmasheath can therefore be modeled from the simple apparatus il-lustrated in Fig. 1. That said, the parameters and the initialconditions in the experiment by Arnaset al. can be summa-rized as follow:

κ2 = 6.5, κ3 = 1, h = 5mm, Ep = 5.15kV/m,σ1 = 0C/m2, σ2 =−5.35µC/m2,m ≈ 15.1ng, a = 0m, b = 32µm,

zd (0) = 4.4mm, zd (0) = 0mm/s, zd ≡ dzd/dt,

(8)

where the details of how these were obtained are explainedin the section B of the appendix. Using these, the equationof motion described in Eq. (5) can be solved numerically viaRunge-Kutta method. The result is shown in Fig. 4. One

5

0123456789

10

0 5 10 15 20

z d [n

m]

t [ps]

(a)

Ep = 0.50 GV/mEp = 0.75 GV/mEp = 1.00 GV/m

0123456789

10

0 5 10 15 20

z d [n

m]

t [ps]

(b)

N = 250N = 275N = 300N = 500

-3

-2

-1

0

1

2

3

0 5 10 15 20

Par

ticle

vel

ocity

[km

/s]

t [ps]

(c)

Ep = 0.50 GV/mEp = 0.75 GV/mEp = 1.00 GV/m

23.17100

200

300

400

500

600665.34

0.1 1 2 3 4 4.9

ν osc

[GH

z]

Ep [GV/m]

(d)

Figure 3: (Color online) Equation (5) is plotted, where in (a) electric field is varied and in (b) ion’s surface charge density σ1 = Nqe/(

4πa2)

isvaried atEp = 1GV/m. (c) Ion velocity corresponding to the plot in (a). (d) Dependence of oscillation frequencyνoscon electric field strengthEp. Similar plot for the lower frequencies involving weakly charged ions subject to smallerEp is also provided in the section E of appendix.In (a,b,c,d), all of the unspecified parameter values are from Eq. (6) and the initial conditions are from Eq. (7).

can readily verify that the particle oscillates at a frequencyof approximatelyνosc≈ 17Hz, which is consistent with themeasurement by Arnaset al.40

Although Arnaset al. explicitly states that their glass mi-crosphere is hollow structured, their paper does not provideany information regarding the radius of the void inside oftheir hollow glass microsphere. Due to the lack of this in-formation, the mass of an hollow glass microsphere in Eq.(8) has been computed assuming a solid microsphere. Suchassumption overestimates the mass of an hollow glass mi-crosphere. In fact, in their paper, Arnaset al. indicates thegravitational force ofFg = (1.5±0.3)×10−10N for their hol-low glass microsphere.40 Such force corresponds to a massof m ≈ 15.3ng, which is exactly the mass computed as-suming a solid glass microsphere in Eq. (8). If instead asmaller mass ofm ≈ 6.19ng is assumed, an electric fieldstrength ofEp = 2.2kV/m is sufficient to generate an oscilla-tion frequency ofνosc≈ 17Hz. This value for the electric fieldstrength, i.e.,Ep = 2.2kV/m, is consistent with the local elec-tric field strength in the plasma sheath obtained by Arnaset al.in their dusty plasma experiment. The particle’s oscillatorymotion corresponding to this latter case, wherem ≈ 15.1ngandEp = 5.15kV/m, has been plotted in Fig. 4 for compari-son.

4.4

4.5

4.6

4.7

4.8

4.9

0 0.05 0.1 0.15 0.2 0.25 0.3

z d [m

m]

t [s]

m = 6.19 ng, Ep = 2.20 kV/mm = 15.1 ng, Ep = 5.15 kV/m

Figure 4: (Color online) Equation (5) is plotted using parameter val-ues and initial conditions defined in Eqs. (8). Oscillation frequencyof νosc≈ 17Hz is predicted by the theory, which is consistent withthe measurement by Arnaset al.40 In the plot, the anode is located atzd = 0mm and the cathode is located atzd = 5mm.

This remarkable result demonstrates that self-sustained os-cillations in both dusty plasmas and DC glow discharges sharea common physical mechanism, in which the oscillations can

6

be attributed to the interaction between the ionized particleand the surface charges induced by it at the bounding elec-trodes. Such outcome makes sense because the oscillatingionized particles in both dusty plasmas and DC glow dis-charges experiments are only differ in their physical sizesandmasses. Apart from this, the two systems are nearly identicalin nature from the physics point of view.

Quite often in dusty plasmas, theoretical models based onsimple harmonic excitations of small amplitude oscillationsare employed for the description of the particle dynamics. Onesuch model is given by41

d2zdt2 +νdn

dzdt

+Ω2vz =

f0 cosωtmd

, (9)

wherez is the particle displacement,md is the particle’s mass,ω is the angular frequency of oscillation,t is the time, andf0 is the amplitude of the external force. Particle’s equationof motion based on such model, Eq. (9), is limited to thedescriptions of oscillations involving a small amplitude dis-placements about the stationary equilibrium position. Besidesthis limitation, the equation of motion described by Eq. (9), asit stands, cannot be directly applied to describe the particle’soscillatory motion due to the fact that the quantitiesνdn andΩv must be first determined experimentally.42 Consequently,Eq. (9) is only useful when determining the total charge car-ried by the particle. Although Eq. (9) provides an indirectmethod of measuring the particle’s total charge, as it stands,it says nothing about the fundamental mechanism behind theself-sustained oscillations of ions in dusty plasmas.

The equation of motion described in Eq. (5) is distin-guished from the one illustrated in Eq. (9) in that its physi-cal description is not limited to just small amplitude oscilla-tions but covers particle oscillations of any amplitude ranges.Further, Eq. (5) does not depend on quantities likeνdn andΩv, which terms must be determined experimentally. Instead,Eq. (5) completely describes the dynamics of charged-particlemotion solely based on the information obtained from theparticle’s physical properties (i.e., mass, surface charge den-sity, and dielectric constant, etc.) and the local electricfieldstrength. In the case where the mass of an ionized particleis represented by that of an ionized atom, Eq. (5) describesthe dynamics of self-oscillating ionized atom in the DC glowdischarges. In the previous work,29 I have discussed that Eq.(5) yields results that qualitatively agree with certain aspectsof self-oscillations in the DC glow corona experiments pre-dicted by theories based on the fluid models.23,25 If plasmain the DC glow corona experiment is effectively treated as aself-oscillating weakly ionized single superparticle, Eq. (5)provides a satisfactory description of the electrode current os-cillations in the DC glow corona. Further discussion on thisisprovided in the section E of the appendix.

Until now, no single theory was able to successfully ex-plain self-sustained oscillations in both dusty plasmas and DCglow discharges experiments. For years, experimental evi-dences hinted that these were related phenomena, but withoutany definitive conclusions. In this respect, Eq. (5) is the firsttheoretical model to provide such definitive conclusions.

IV. DEVICE APPLICATION

It is well known that oscillating ions generate electric dipoleradiation. Such property can be utilized to develop a widebandelectromagnetic radiation source in which the frequency ofemitted radiation can be tuned by varying the DC bias voltageacross the electrodes.27,29 According to Zouche and Lefort,the plane-parallel plate electrodes made of nickel-silvercom-posite material, which are separated by a gap ofh = 100nm,can support DC bias voltages up to∼ 400V across the elec-trodes before an onset of electrical breakdown.43 With im-provements in the electrical breakdown characteristics, suchdevice can be engineered to cover both the microwave andthe terahertz band of electromagnetic spectrum with high effi-ciency.

The first experimental evidence that the phenomenon ofself-sustained ion oscillations in the DC glow discharges has apotential applications in the development of a wideband elec-tromagnetic radiation source came from McClure in 1963,when he reported an oscillation frequency ofνosc≈ 20MHz ina low pressure glow discharge tube, which comprised of cylin-drical hollow cathode and a very thin coaxial wire anode.44

Unfortunately, McClure provided no theoretical model to ex-plain his finding. Such concentric cylinder configuration forthe electrodes is a typical apparatus found in many DC glowdischarges experiments. Nearly two decades later, in 1980,Alexeff and Dyer shrunk the aforementioned concentric cylin-der configuration (filled with air at gas pressure of 0.1mTorr)to the size of a pen and successfully demonstrated the gen-eration of microwave radiations in the gigahertz frequencyranges.45 Their pen sized coaxial configuration was later re-ported to generate radiation at terahertz frequency ofνosc≈

1THz with an output radiation power ofPrad ≈ 1.5W.46,47 Inorder to explain their observations, Alexeff and Dyer proposeda model which they referred to as theorbitron theory.45–47Thebasic idea behind the orbitron theory is very simple. The elec-trons emitted from the inner surface of the outer concentriccylinder (which part represents the cathode in the coaxial con-figuration) orbit around a thin coaxial wire anode; and, suchelectron orbital motion generates the detected high frequencyelectromagnetic radiation. However, various experimental re-sults reported by other groups48–50 contradicted the orbitronmodel; and, the orbitron theory is no longer considered as thecorrect description of the physics behind the observationsre-ported by Alexeff and Dyer.

Somewhat similar to the aforementioned pen sized coax-ial configuration investigated by Alexeff and Dyer is the mi-crohollow cathode discharges (MHCDs) configuration, whichwas first introduced by Schoenbachet al.13 However, unlikethe coaxial configuration, where the self-pulsing frequen-cies of ions typically lie in the gigahertz ranges, the self-pulsing of ions in the DC glow discharges for typical MHCDsconfigurations lie only in some tens of kilohertz frequencyranges.21,51,52Why? The answer to such discrepancy lies inthe differences in the gas pressures applied in two configura-tions. In the high frequency design by Alexeff and Dyer, thepen sized coaxial device is filled with gas at a very low gaspressure of 0.1mTorr whereas, in typical MHCDs configura-

7

tions, the device is maintained at gas pressures on the orderoftens (or hundreds) of torrs. For instance, in the MHCDs ex-periment by Aubertet al.,51,52the gas pressure ranged from 40to 200Torr. Heet al.21 worked with somewhat lower gas pres-sure of∼1Torr (i.e., 133Pa) for their MHCDs experiment, butcompared to the gas pressure of 0.1mTorr used by Alexeff andDyer in their pen sized coaxial configuration, this is still largerby a factor of ten thousand. Consequently, in the aforemen-tioned MHCDs experiments, the DC glow discharges involvemuch weaker static electric fields due to larger screening ef-fects. When a gas filled medium is applied with an externalstatic electric field, the polarization process sets in and suchscreening effect weakens the net electric field strength in theregion filled with gas. The degree of such screening processgrows with gas pressure. Consequently, in typical MHCDsexperiments, where much higher gas pressures are involved,relatively low self-pulsing frequencies are observed as a re-sult of larger screening effects for the applied static electricfield. Contrary to this, in the aforementioned pen sized coax-ial configuration investigated by Alexeff and Dyer, the exter-nally applied static electric field is only weakly screened dueto the fact that the device is maintained at a very low gas pres-sures. Consequently, the ions in such device can oscillate atvery high frequencies.

Besides this screening effect which acts to weaken the ex-ternally applied static electric field, the presence of higher gaspressure increases the effective mass of an oscillating super-particle. The superparticle concept has been previously intro-duced in the discussion of self-oscillations involving a plasmafireball. The dynamics of plasma involves collective motionsof its constituent atoms. In the framework of this alternativetheory27,29elaborated in this paper, the self-pulsing dynamicsin both MHCDs experiments and the pen sized coaxial con-figuration investigated by Alexeff and Dyer involves the con-cept of oscillating superparticles. The effective masses of suchsuperparticles have an explicit dependence on gas pressures.In general, the superparticle associated with the higher gaspressure environment has a larger mass than the one associ-ated with the lower gas pressure environment. Consequently,the superparticle in a typical MHCDs experiment has muchlarger mass than the one in the pen sized coaxial configurationinvestigated by Alexeff and Dyer. This also explains why self-pulsing frequencies are much lower in MHCDs experimentscompared to the device investigated by Alexeff and Dyer.

V. CONCLUDING REMARKS

In summary, the mechanism behind self-sustained oscilla-tions of an ion in the DC glow discharges has been briefly dis-cussed in the framework of interaction between an ion and sur-face charges that it induces at the bounding electrodes. Suchalternative description provides an elegant explanation to theformation of plasma fireballs in the laboratory plasma. Ithas been found that oscillation frequencies also increase withion’s surface charge density, but at the rate which is signifi-cantly slower than it does with electric field strength. Suchresult supports the conclusions by Fox3 and Bošanet al.,36

where they reported of oscillation frequencies being not toodependent on the type of ions used in the discharge. Self-sustained oscillations in the DC glow discharges can be ex-pected to have high thermal stability due to ion’s kinetic en-ergies that are much larger than the average room tempera-ture thermal energies. It is well known that self-sustainedoscillations in the DC glow discharges occur only in thenegative resistance region of the voltage-current characteris-tic curve. Experimentally, such region is characterized byVth1≤VT ≤Vth2, whereVth1 andVth2 are some threshold volt-ages. Such observation is quite elegantly explained from thesolutions of Eq. (5), where the physical solutions are onlyfound for Vth1 ≤ VT ≤ Vth2. Presented mechanism also cor-rectly describes the self-sustained oscillations of ions in dustyplasmas. To demonstrate this, Eq. (5) has been applied tocorrectly predict the frequency of dust particle oscillationsin the dusty plasmas experiment by Arnaset al.40 Such re-sult demonstrates that self-sustained oscillations in dusty plas-mas and DC glow discharges involve common physical pro-cesses. This is the first theory to successfully explain the self-sustained oscillations phenomena in both dusty plasmas andthe DC glow corona physics.

APPENDIX

A. Sheath potential

It is worthwhile to compare the potential energy well ofFig. 2 with the empirical potential well introduced by Tommeet al.34 to describe the potential in the plasma sheath. Theplasma sheath is an empty space residing between the plasmaand the confining electrodes. One such plasma sheath nearthe anode is illustrated in Fig. 5(a). For the reason that elec-trons and ions move at different velocities due to the differ-ence in their masses, plasmas are never completely neutral atany instant. Consequently, a plasma confined between DCvoltage biased electrodes effectively behaves as if it wereasuper large charged-particle. For brevity, such “super largecharged-particle” shall be simply referred to as a “plasmaball” throughout the discussion here. In fact, the single ion-ized nanoparticle case illustrated in Fig. 5(b) is the limitinwhich the plasma in Fig. 5(a) reduces down to contain justone ionized nanoparticle. That said, just as single ionizednanoparticle self-oscillates when confined by DC voltage bi-ased electrodes, the plasma ball also self-oscillates betweenthe anode and cathode electrodes.9 However, due to its largemass, the plasma ball oscillates at much lower frequenciescompared to the nanoparticle counterparts. For instance, as-suming the gaph between the plate electrodes is in the orderof sub-millimeters or so in Fig. 5(a), the macroscopic plasmaconfined between such electrodes can contain very large num-ber, i.e., say, millions or more, of ionized nanoparticles oratoms depending on the gas pressure. In general, for the studyof self-sustained oscillations, the plasma ball illustrated in Fig.5(a) can be effectively modeled using an ionized superparticle,where the terminology superparticle refers to a particle whichis extremely large and enormously heavier compared to the

8

ionized nanoparticle (or atom) illustrated in Fig. 5(b). Insuchmodel, the potential in the plasma sheath is represented bythe one presented in this article. It is quite remarkable thatthe sheath potential introduced empirically by Tommeet al.closely resembles the potential described in this work.

(a) (b)

VL

VT VT

VL

3κ

Ep

Ep

3κAnode

Cathode Cathode

Anode

Sheath

Plasma

Sheath

Ionized nanoparticle

zd zd

0.5h

h

Figure 5: (Color online) (a) Illustration of the plasma and the sheath.The plasma ball effectively behaves as one very large charged super-particle. In the illustration, the plasma has been deliberately drawn asa sphere to emphasize a plasma ball. In reality, however, theplasmacan be in any shape depending on the geometry of confining elec-trodes. (b) The single ionized particle configuration considered inthis article. In (a) and (b), only the sheath near the anode isshownfor illustration.

B. Parameters in experiment by Arnaset al.

Arnaset al.40 reported that a dust particle made of hollowglass microsphere of radiusr ≈ 32µm with mass densityρm ≈

110kg/m3 and carrying a total charge ofQ ≈ −4.3×105qe,whereqe = 1.602× 10−19C, self-oscillates at frequency ofνosc ≈ 17Hz inside the plasma sheath. The inner environ-ment of the plasma sheath, in many respects, is similar to theenvironment in an empty space between the DC voltage bi-ased plane-parallel conductors. In that regard, the undampedself-sustained oscillations of charged glass microspherein theexperiment by Arnaset al. should be theoretically describ-able using the model presented in this article, where a core-shell structured charged particle is confined between the DCvoltage biased plane-parallel conductors. The glass micro-sphere used by Arnaset al. is not core-shell structured, ofcourse. Such dielectric particle is obtained in the limit theradius of conductive core portion of the core-shell structuredparticle goes to zero, i.e.,a = 0. Consequently,σ1 also van-ishes in that limit. That said, the glass microsphere of radiusb = 32µm with mass densityρm ≈ 110kg/m3 has total massof m = (4/3)πb3ρm or m ≈ 1.51×10−11kg. Although Arnaset al. specifically uses the word “hollow glass microsphere”in their report, they do not provide any physical details of theparticle other than its outer radius. Thus, in the aforemen-tioned calculation of the particle’s mass, I have assumed asolid glass microsphere. The glass microsphere carries a to-tal charge ofQ ≈−4.3×105qe, whereqe = 1.602×10−19C.The surface charge density at the radiusr = b is hence given

by σ2 = Q/(

4πb2)

or σ2 = −5.35µC/m2. Arnaset al. doesnot provide any information regarding the dielectric constantκ2 for their glass microsphere. Therefore,κ2 = 6.5 has beenchosen for the dielectric constant of glass microsphere, whichvalue is typical of glass microspheres. For convenience, itshall be assumed that the medium in which the glass micro-sphere oscillates is a vacuum, i.e.,κ3 = 1. The parameters inthe experiment by Arnaset al. is summarized here for refer-ence:

κ2 = 6.5, κ3 = 1, h = 5mm, Ep = 5.15kV/m,σ1 = 0C/m2, σ2 =−5.35µC/m2,m ≈ 15.1ng, a = 0m, b = 32µm,

(10)

where the mass is in nanograms, i.e.,m ≈ 1.51×10−11kg=15.1ng. The initial conditions,

zd (0) = 4.4mm anddzd (0)

dt= 0

mms

, (11)

have been chosen purely out of convenience. With Eqs. (10)and (11), the equation of motion defined in Eq. (5) can besolved numerically via Runge-Kutta method to obtain the re-sults illustrated in Fig. 4. One can readily verify that the the-ory agrees with the experiment.40

C. Comparison to the potential energy of negatively chargedglass microsphere measured by Arnaset al.

According to the model elaborated here,29 the positivelycharged particle confined between a DC voltage biased plane-parallel conductors results in the formation of potential energywell in vicinity of the anode whereas a negatively charged par-ticle results in the formation of potential energy well in vicin-ity of the cathode. Arnaset al.35 have experimentally verifiedsuch potential energy well for the case of negatively chargedparticle in the plasma sheath near the cathode. As explainedpreviously, the problem of charged particle inside the plasmasheath can be effectively modeled by a charged particle con-fined by the DC voltage biased plane-parallel conductors. Thepotential energy functionU (zd) of Eq. (4) has been plotted forthe following parameter values:

κ2 = 6.5, κ3 = 1, h = 5mm, Ep = 3.15kV/m,σ1 = 0C/m2, σ2 =−5.795µC/m2,

m ≈ 4.91×10−12kg, a = 0m, b = 22µm.(12)

The result is shown in Fig. 6, where it shows the formation ofpotential energy well in vicinity of the cathode. One noticesthat the order of magnitude for the potential energy well iscomparable to the experiment by Arnaset al. The minor dis-crepancies in the potential energy well of Fig. 6 and the onemeasured by Arnaset al.35 can be attributed to the fact that theelectric field is not constant in the plasma sheath whereas, be-tween the DC voltage biased plane-parallel conductors, elec-tric field is a constant.

9

-16

-14

-12

-10

-8

-6

-4

-2

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8

U [1

0-15 J

]

zd [mm]

Figure 6: The potential energy of Eq. (4) is plotted for parametervalues defined in Eq. (12). In the plot, the anode is located atzd =0mm and the cathode is located atzd = 5mm. This result should becompared with the potential energy measured by Arnaset al.35 intheir dusty plasmas experiment. One can verify that two results areremarkably similar in potential energy profile as well as in the orderof magnitudes.

D. Physical and unphysical oscillatory solutions

The oscillatory solutions obtained forEp . 0.1GV/m andEp & 4.9GV/m are unphysical in Fig. 3(d). To demon-strate this, Eq. (5) is numerically solved and plotted forEp . 0.1GV/m andEp & 4.9GV/m using the following ini-tial conditions and parameter values:

κ2 = ∞, κ3 = 1, h = 100nm, VL = 0V,σ1 ≈ 3.19C/m2, σ2 = 0C/m2,

m ≈ 8.95×10−24kg, b = a = 1nm,zd (0) = 0.5nm, zd (0) = 0nm/s, zd ≡ dzd/dt.

(13)

To show thatEp . 0.1GV/m yields unphysical solutions,zd (t) is plotted forEp = 0.1GV/m andEp = 0.09GV/m. Theresults are shown in Fig. 7, where the surface of anode is atzd = 0nm, the surface of cathode is atzd = 100nm, and themidway between the plates is atzd = 50nm. It can be clearlyseen that forEp = 0.09GV/m, positively ionized particle peri-odically crosses the midway between the two electrodes. Suchcase is unphysical because, for positively ionized particles,oscillations can only exist forzd < h/2− b.27 Now, it can beverified that anyEp less thanEp = 0.09GV/m yields suchunphysical solutions.

To show thatEp & 4.9GV/m yields unphysical solutions,zd (t) is plotted forEp = 4.9GV/m andEp = 5.0GV/m. Theresults are shown in Fig. 8(a), where the plot has been en-larged for a view nearzd = 0pm in Fig. 8(b). It can beclearly seen that forEp = 5.0GV/m, positively ionized par-ticle periodically penetrates into the surface of anode, whichis unphysical. Now, it can be verified that anyEp larger thanEp = 5.0GV/m yields such unphysical solutions.

For the initial conditions and parameter values defined inEq. (13), physical solutions for the oscillatory motion of pos-itively ionized particle are only found forEp satisfying thecondition given by 0.1GV/m . Ep . 4.9GV/m, where the

0

10

20

30

40

50

60

0 20 40 60 80 100 120 140

z d [n

m]

t [ps]

Ep = 0.09 GV/mEp = 0.10 GV/m

Figure 7: (Color online) Plot ofzd (t) for Ep = 0.1GV/m andEp = 0.09GV/m. The surface of anode is atzd = 0nm, the surfaceof cathode is atzd = 100nm, and the midway between the two elec-trodes is atzd = 50nm.

-500

100

200

300

400

500

0 1 2 3 4 5

z d [p

m]

t [ps]

(a)

Ep = 4.9 GV/mEp = 5.0 GV/m

-50

0

50

100

0 1 2 3 4 5

z d [p

m]

t [ps]

(b)

Ep = 4.9 GV/mEp = 5.0 GV/m

Figure 8: (Color online) (a) Plot ofzd (t) for Ep = 4.9GV/m andEp = 5.0GV/m. (b) The plot in (a) has been enlarged for a viewnearzd = 0pm. The surface of anode is atzd = 0pm.

upper and lower bounds ofEp are rough estimates based onthe grounds of the discussed physicality. This result explainswhy oscillations suddenly appear at certain “initial” thresholdDC bias voltage and disappear suddenly when bias voltagegoes beyond certain larger “final” threshold voltage in exper-iments.

10

E. Weakly ionized particle confined between anode andcathode plates separated by gap ofh = 1mm

The oscillation frequency plot of Fig. 3(d) has been ob-tained for an highly ionized particle which is confined be-tween two plate electrodes with very small separation gap ofh = 100nm. Here, the same calculation is done for a weaklyionized particle confined between plate electrodes with mi-croscopically very large separation gap ofh = 1mm. Ac-cording to Chenet al.,32 a spherical argon cluster of radiusr = a = 4.7nm contains roughly∼ 9600 argon atoms at back-ing pressure of 50 bars. Since the particle is assumed to beweakly ionized, I shall assume thatN = 20 electrons are re-moved from it. This corresponds to surface charge densityof σ1 = Nqe/

(

4πa2)

or σ1 ≈ 1.1543× 10−2C/m2, whereqe = 1.602×10−19C is the fundamental charge unit. Neglect-ing the masses of missingN electrons, the particle has a totalmass ofm≈ 6.365×10−22kg. For the calculation of particle’smass, it has been assumed that single argon atom has massof 6.63×10−26kg. That said, the following initial conditionsand parameter values are used for the calculation of particle’soscillation frequency as function of electric field strength:

κ2 = ∞, κ3 = 1, h = 1mm, VL = 0V,σ1 ≈ 11.543mC/m2, σ2 = 0C/m2,

m ≈ 6.365×10−22kg, b = a = 4.7nm,zd (0) = 10µm, zd (0) = 0µm/s, zd ≡ dzd/dt.

(14)

Using these, the equation of motion defined in Eq. (5) issolved numerically via Runge-Kutta method for the oscilla-tion frequency as function of electric field strength; and, theresult is shown in Fig. 9.

This result should be compared with the one provided byAkishev et al.,25 where they have plotted both experimen-tal and calculated periodTosc of self-sustained oscillationsagainst the average corona current at different pressures ofnitrogen. Fox3 and Bošanet al.36 have shown that oscilla-tion frequencies are not very dependent on the type of ionizedparticles used in the discharge. That said, the result obtainedby Akishevet al. for nitrogen gas can be compared with thecalculation done here for ionized spherical argon cluster.Re-calling thatνosc= 1/Tosc and the average corona current inthe electrode increases with electric field strength, it canbereadily convinced that the profile of result shown in Fig. 9 isconsistent with the result obtained by Akishevet al.

Although the profile of oscillation frequency dependenceon electric field strength (or corona current in the electrode)is comparable in both results, the measurement by Akishevetal. shows much lower oscillation frequencies for given rangeof electric field strengths. Why? Such discrepancy arisesfrom the fact that in the experiment by Akishevet al., theself-oscillating object is a charged plasma ball, i.e., chargedsuperparticle, whereas in the calculation of Fig. 9, the self-oscillating object is a charged nanoparticle. As already dis-cussed in section A, a macroscopic plasma ball contains verylarge number of ionized nanoparticles (or atoms) dependingon the gas pressure. This makes macroscopic plasma ball ef-

fectively a single charged superparticle with very large mass.

0

50

100

150

200

250

300

350

0 0.5 1 1.5 2 2.5 3 3.5

ν osc

[kH

z]

Ep [kV/m]

(a)

7.620

40

60

80

100

120

0.01 0.1 0.2 0.3 0.4 0.5

ν osc

[kH

z]

Ep [kV/m]

(b)

Figure 9: (a) The frequency of oscillations is plotted against electricfield strength for a weakly ionized particle with initial conditions andparameters values defined in Eq. (14). (b) The plot in (a) is enlargednear zero. Unlike the case illustrated in Fig. 3(d), where ionizedparticle is confined between electrodes with very small gap,the re-sult here involves very large number of discretized time steps forthe Runge-Kutta routine. For that reason, the oscillation frequencyhas been plotted fromEp = 0.01kV/m toEp = 3.5kV/m in (a). Thethreshold electric field strengths, i.e.,Ep,th1 andEp,th2, are not shownin the figure.

It has been illustrated in Fig. 4 that an ionized particle withsmaller mass requires weaker electric field strength comparedto the one with larger mass to oscillate at the same frequency.Mathematically, such characteristic arises from the fact thatEq. (5) has mass dependence in the denominator. Conse-quently, the plasma ball in the experiment by Akishevet al.,which effectively behaves as a single charged superparticle,oscillates at much lower frequencies for the given range ofelectric field strengths compared to the configuration used inFig. 9, where the mass of an ionized particle is much smallerthan the effective mass of a plasma ball. In principle, once themass and the effective total charge information of the plasmaball, i.e., superparticle, are provided, the oscillation frequencydependence on corona current in the electrode (or electric fieldstrength) measured by Akishevet al. can be reproduced by thepresented theory.

11

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