Inductively coupled plasmas at low driving frequencies - DOE …doeplasma.eecs.umich.edu/files/PSC_Kolobov6.pdf · 2018. 2. 2. · Inductively coupled plasmas at low driving frequencies

Inductively coupled plasmas at low drivingfrequencies

Vladimir I Kolobov1 and Valery A Godyak2

1CFD Research Corporation, University of Alabama, Huntsville, AL, United States of America2 RF Plasma Consulting and University of Michigan, Brookline, MA, United States of America

E-mail: [email protected]

Received 13 October 2016, revised 14 May 2017Accepted for publication 30 May 2017Published 5 July 2017

AbstractWe discuss the peculiarities of inductively coupled plasma (ICP) at low driving frequencies. Theratio of electric to magnetic field, E cB ,∣ ( )∣ decreases with decreasing frequency according toFaraday’s law—higher magnetic fields are required to induce the same electric field at lowerfrequencies. We point out that the ratio of E cB∣ ( )∣ can be non-uniform in space depending onprimary coil configuration and the presence of ferromagnetic materials. In this paper, weconsider examples of low-frequency ICPs with negligibly small magnetic fields in plasma. Thedisparity of time scales for ion transport and the electron energy relaxation results in nonlinearplasma dynamics at low frequencies. Numerical simulations demonstrate that at low frequencies,the presence of plasma has very little effect on spatial distributions of the electric and magneticfields, which are determined solely by the coil geometry and by the presence of ferromagneticcores. Simulations of plasma dynamics in ICP over a wide range of driving frequencies and gaspressures illustrate high-frequency, quasi-static and dynamic regimes of discharge operation andexplain some trends observed in experiments.

Keywords: Inductively coupled plasma, magnetic fields, low frequency, ferromagnetic, skin effect

1. Introduction

Inductively coupled plasma (ICP) can be maintained over awide range of driving frequencies from 50 Hz up to GHz[1–3]. Most ICPs used for ion sources and material processingoperate at relatively high frequencies f = 2–27MHz when theangular driving frequency ω = 2πf is comparable and exceedsthe electron collision frequency with gas species, ν. Underthese conditions, electron temperature and plasma density donot oscillate over the RF period, the plasma electricalresponse is linear, and the electron heating process is deter-mined by an effective electric field. At these frequencies, dueto the skin effect, electromagnetic fields are localized within askin layer near the plasma boundary adjacent to the inductor.

In this paper, we identify the most important aspects ofICP behavior at low frequencies:

(a) The spatial distributions of the electric and magneticfields depend on ICP geometry and the presence offerromagnetic materials. For some configurations,substantial electric field can be induced in the areaswith negligibly small magnetic field.

(b) Three operating regimes can be distinguished depend-ing of the field frequency and the characteristic timescales for electron energy relaxation and ion transport.

(c) With decreasing frequency, the skin effect disappears,and the spatial distributions of electromagnetic fieldsare not affected by the presence of plasma and aredetermined solely by the geometry of the system.

We first identify characteristic time scales and distinguishquasi-static, high-frequency and dynamic regimes of dis-charge operation depending on the value of driving frequencywith respect to the characteristic time scales. We discussdistributions of electric and magnetic fields and give exam-ples of ICP with negligibly small B field in the plasma. Self-consistent 2D simulations illustrate the effects of primary coilconfiguration and the presence of ferromagnetic cores onspatial distributions of electromagnetic fields. Then, we pre-sent the results of simulations for plasma dynamics in ICPwith a closed ferrite core over a wide range of frequencies forargon gas. A fluid model is used to illustrate the peculiar effectsobserved at different frequencies and the specifics of thedynamic regime of plasma operation. As an example of kinetic

Plasma Sources Science and Technology

Plasma Sources Sci. Technol. 26 (2017) 075013 (13pp) https://doi.org/10.1088/1361-6595/aa7584

0963-0252/17/075013+13$33.00 © 2017 IOP Publishing Ltd1

effects, we demonstrate dynamic constriction of the plasmacolumn observed in simulations with non-Maxwellian EEDF.

2. Specifics of ICP at low driving frequency

In this section, we first introduce the characteristic time scalesand distinguish three operating regimes (quasi-static, high-frequency and dynamic). Then, we discuss the importance oftime-varying magnetic fields for ICPs and consider certainICP configurations with negligibly small magnetic fields inplasma.

2.1. Time scales and operating regimes

Let us limit ourselves to discharges in noble gases at rela-tively low gas pressures, when the loss of charged particles isdetermined by surface recombination rather than volumerecombination, and there is no electron attachment. In thiscase, the largest characteristic time scale is the time τa ofplasma transport to the wall. For a highly collisional plasma,the corresponding characteristic frequency νa is determinedby the ambipolar diffusion, νa = Da/Λ

2, where Λ is thecharacteristic size of the plasma and Da is the ambipolardiffusion coefficient. For nearly collisionless plasma, νa =vs/Λ where vs is the ion sound speed.

The second characteristic frequency νε corresponds to theelectron energy relaxation scale. The latter is defined by thecharacteristic times of electron heating by the electric field, andthe characteristic time of electron energy loss in collisions withatoms and Coulomb interactions among electrons. The heatingand cooling rates are equal to each other in steady plasma butcan differ substantially in transient plasmas. The characteristicfrequency of electron heating by the electric field is noted as .h

The characteristic frequency of electron energy relaxation incollisions, w w w w ,c

ee*( ) ( ) ( ) ( ) depends onthe electron kinetic energy w and on the plasma density. Here,

is the relative energy lost in elastic collisions with atoms,w*( ) is the total inelastic collision frequency (including

excitation and ionization), and wee( ) is the frequency ofCoulomb interactions among electrons. We can define as

max ,h c{ } and notice that for weakly ionized plasmaa due to the large difference between electron and

ion mass.Having defined the frequencies νa and νε, we can dis-

tinguish three operating regimes depending on the values ofthe driving frequency ω and these frequencies. The bound-aries of quasi-static, dynamic and high frequency regimescorrespond to the conditions when ω becomes equal to thefrequencies νa and νε, which correspond to the characteristictime scales of the ion transport to the wall τa = a

1 and theelectron energy relaxation time τε = .1

The quasi-static regime corresponds to a low-frequencycase a( ) when plasma density varies significantly overthe field period following the electric current, and the electricfield and electron temperature exhibits highly nonlinearbehavior. We show below that the electric field and electrontemperature remain practically constant during the most part

of the half-period and drop sharply near current zero. Suchcurrent and voltage waveforms are typical for fluorescentlamps driven at line frequency 60 Hz [4].

At high frequencies ( ) the electron energy dis-tribution function (EEDF), f0(ε), and the electron temperature,Te, are constant over the period, and are determined by theelectric field, E E 1 ,0

2eff2 1 2( ) where E0 is the elec-

tric field in DC plasma at the same gas pressure and geometry,and νeff is the effective electron collision frequency. The fieldE0 depends on the product pΛ, and adjusts itself according toelectron energy balance [5].

The intermediate case (νa < ω < νε) corresponds to a‘dynamic regime’ [6]. In this regime, the plasma densityvaries slightly over the field period but the electron temper-ature, ionization rate, and the EEDF shape could changesignificantly over the field period. The dynamic regime occursin fluorescent lamps driven at a frequency of 20–100 kHz [7].No detailed studies of this regime have been performed so far,and the present paper, to our knowledge, is the first step inthis direction.

2.2. Electromagnetics

It is well known that ICP is maintained by an inductiveelectric field, E, which according to Faraday’s law, is pro-duced by a time-varying magnetic field, B. However, effectsof B field on ICP properties have rarely been discussed. Here,we point out that the ratio E cB∣ ( )∣ can be non-uniform inspace. For some ICP configurations, substantial electric fieldcan be induced in the areas with negligibly small magn-etic field.

To illustrate this apparently paradoxical and counter-intuitive situation, let us consider a long solenoid, or a thintorus with a poloidal surface current. In a static case, themagnetic field is confined inside the solenoid or torus and isabsent outside. In a quasi-static case, at slow time variationsof the current, the induced electric field is present both insideand outside the solenoid or torus, but the magnetic fieldoutside is very small and is due to the displacement currentonly. In quasi-magnetostatic (QMS) approximation (seebelow), the magnetic field outside is zero, so the ratio ofE cB∣ ( )∣ jumps on the surface where the current flows.Therefore, if plasma is created inside the solenoid or torus, itwill be embedded into the B field, whereas plasma createdoutside will be free from the inductor’s magnetic field; onlythe magnetic field created by the plasma current will bepresent there.

Furthermore, let us consider how the magnitudes of theelectric and magnetic fields change with driving frequency.For ω < ν, the electric field required to maintain plasmadepends weakly on frequency. Since E ∼ ωB, the magnitudeof the B field needed to maintain plasma varies with fre-quency as B ∼ ω−1. If the time-varying magnetic field ispresent in plasma, electrons can be magnetized over a part ofthe field period and de-magnetized when the B-field is nearzero. The large AC magnetic field at low driving frequenciesmay significantly affect plasma transport and plasma elec-trodynamics, giving rise to nonlinear effects due to the

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Plasma Sources Sci. Technol. 26 (2017) 075013 V Kolobov and V Godyak

Lorentz force [8–16]. The existence of the second harmoniccurrent flowing normal to the main discharge current at afundamental frequency, and the ponderomotive effect sig-nificantly changing the spatial distribution of plasma densitywere found in ICP operating at ω/2π = 0.4 MHz [12, 14, 15].However, most of the previous papers considered the effectsof the magnetic field on electron transport in terms of pon-deromotive forces. The effects of time-varying magneticfields on electron kinetics and plasma transport in low-fre-quency ICPs have not been discussed in detail, even for thesimplest cases.

In this paper, we consider ICPs with negligibly smallmagnetic fields in plasma. Examples of such systems are a so-called ferromagnetic enhanced ICP, FMICP, or transformerICP. The term transformer ICP denotes an ICP enhanced witha ferromagnetic core of high permeability forming a closedmagnetic circuit [17–20]. In such ICPs, the magnetic fieldproduced by the primary current, is confined inside a magn-etic path loop outside the plasma. An example of an oppositecase with a strong AC magnetic field in plasma due to theprimary current is a spherical theta pinch [21].

2.3. Skin effect, power coupling, and conditions for a stabledischarge operation

The penetration of electromagnetic fields into plasma ischaracterized by a skin effect. The thickness of the skin layer,δ, increases with decreasing field frequency. At low fre-quencies, when δ � Λ, plasma has little effect on the spatialdistributions of the electromagnetic fields, which are deter-mined solely by the system geometry and magnetic propertiesof materials. However, the mechanisms of power transferfrom the electromagnetic fields to plasma electrons are highlysensitive to spatial distributions of the fields in plasma. Col-lisionless electron heating associated with thermal motion ofelectrons in spatially inhomogeneous electromagnetic fieldstakes place under conditions when v ,e 0 [12]. Here 0

is the characteristic scale of the electric field non-uniformitydue to geometrical factors, E E ,0 ∣ ∣ ∣ ∣ and ve is theelectron thermal velocity.

As a conventional transformer, ICP provides magneticcoupling between primary and secondary circuits. Themagnetic flux induced by plasma current interacts with theprimary flux induced by the coil current independently on thepresence or absence of skin effect. In the case of a skin effect(δ < Λ), the last can (partially or totally) offset the negativeresistance of plasma by reducing the plasma current crosssection and increasing the electric field in the skin layer. Thiseffect results in an increase of the plasma ballasting, thusaffecting the ICP stability. In the case of δ > Λ, the ICPballasting is only provided by leakage inductance and by theresistance of the power source.

In our calculations described below, we used simplifyingassumptions to avoid modeling the full primary circuit (thataccounts for matcher network and the power source) and toavoid full 3D simulations. In the simulations described insection 3.2, we adjusted the coil current to provide the electricfield required by ionization and energy balances for plasma

stability. In section 4, we did not solve the Maxwell equationsbut specified the discharge current, and calculated the alter-nating electric field from Ohm’s law. The validity of theseapproximations is discussed in sections 3.3 and 4.

3. Spatial distributions of electromagnetic fieldsin ICP

It is well known that a time varying magnetic field induceselectric field (according to Faraday’s law) and a time varyingelectric field induced magnetic field (via displacement cur-rent). For slow time variations, in quasi-static regimes, onecan neglect either the displacement current (quasi-magnetostatics(QMS)), or the magnetic field (quasi-electrostatics (QES)). Theapplicability conditions for both cases are defined by thesmallness of the ratio c, where is the characteristic spatialscale [22]. The QES model describes capatitively coupledplasmas where the displacement current is important andmagnetic fields can be neglected. The QMS model describesinductive effects in ICP where magnetic fields produced byslowly varying conductive currents are much greater than themagnetic fields produced by the displacement currents.

For analysis of electromagnetic fields in low-frequencyICP, we neglect the displacement current and use the QMSmodel (SI units) [22]:

H j, 1( )

tE

B, 2( )

B 0. 3( )Here E and H are the electric and magnetic fields, respec-tively, B H is the magnetic induction, j is the currentdensity, and is the magnetic permeability. Equation (2) tellsus that the electric field E is irrotational where B = 0 but doesnot give any explicit expression for the electric field. In manypractical cases, it is possible to find topologies with non-zeroE but zero B in a workable volume of space where plasma canbe produced. Examples of such systems and limitations of theQMS model are discussed below.

To analyze electromagnetic fields at low frequencies, it isconvenient to use the vector magnetic potential A, which isdefined as

A B 4( )and satisfies the Coulomb gauge

A 0. 5( )By comparing equations (1) and (5) one can see that B is thesource for A just as j is the source of B. This symmetry isquite useful for analysis of field distributions for differenttopologies [23]. Of particular interest to the present paper arecurrent distributions that can produce B = 0 but A 0 in aworkable volume of space where plasma can be created. Thesimplest example of such a system is a long solenoid. Themagnetic field of the solenoid is confined inside the solenoid,whereas the vector magnetic potential and the induced electricfield are present both inside and outside the solenoid. Indeed,

3


small magnetic field proportional to the displacement currentdoes exist outside of the solenoid [22]. However, this field isnegligibly small for the quasi-static case we consider.

Another configuration producing B = 0 but A 0 in aworkable space volume is a torus with a poloidal surfacecurrent. For a steady current, B = 0 but A 0 outside thetorus. For a slowly varying current, B 0 outside within theQMS model, and is very small with account for the dis-placement current [23].

The induced electric field can be calculated from thevector magnetic potential A as

tE

A. 6( )

Thus, it is the local time-dependent A that generates E, notthe local B field.

The fields penetrate into conductors within a skin layer ofthickness . The thickness of the skin layer can be differentfor electric and magnetic fields [24]. At low frequencies,

,1 and the presence of a strong magnetic field in the

skin layer leads to several nonlinear effects in metals [25] andin plasmas [26, 27]. With decreasing frequency, this non-linearity increases. For gas discharge plasma, additionalnonlinearity appears due to the strong dependence of plasmaconductivity on the electric field strength, which is caused byionization processes.

Assuming Ohm’s law for the current density, j E,the vector magnetic potential A can be found from theequation:

tA

Aj

1, 7s ( )

where js is the source current density and σ is the electricalconductivity. When regions with different (piece-wise con-stant) permeability, ,0 r are present equation (7) isreduced to the magnetic diffusion equation [28]:

t

AA j . 8s ( )

Here, μ0 and μr are the permeability of free space and therelative permeability of media, respectively. The boundaryconditions at interfaces of domains with a different relativepermeability r express continuity of all three components ofA, continuity of the normal component of B, and continuity ofthe tangential component of H.

It follows from equation (8) that the spatial distribution ofA (and thus the B and E fields) depends not only on thecurrent source js but also on spatial distributions of the con-ducting and magnetic materials in the system. For analysis ofsystems containing ferromagnetic components that almostentirely confine the magnetic flux, it is convenient to employthe magnetic circuit concept. Taking into account thatB H and j E, the magnetic circuit is analogous to theelectric (resistive) circuit with magnetic permeability rplaying the role of the electric conductivity . The design ofelectrical transformers, where time-varying currents aretransformed from primary to secondary currents, can be wellunderstood using this analogy. For example, by inserting aferromagnetic core inside a solenoidal coil of finite length andby closing the magnetic circuit using a closed core, one candecrease the source current density to obtain the sameelectromagnetic fields in the surrounding air. In practice, thefield values and their spatial distributions depend on aneffective μeff, which is the ratio of the coil inductance withand without a ferromagnetic core, and is a function of r andthe system topology. Below, we discuss specific examples ofsome topologies relevant to practical ICP systems.

3.1. The fields of a solenoid and effects of ferrite cores

To illustrate one of the key points of our paper, let us considerelectromagnetic fields produced by a solenoid of radius a anda length h driven by a sinusoidal current of amplitude I0 andangular frequency . This geometry corresponds to a con-figuration typical of rf compact fluorescent light sources (seefigure 1) driven by the primary coil wrapped around acylindrical tube with N turns per unit length.

Figure 1. The lines of B and a contour of A (red line) for air core(μr = 1) (a), short ferrite core with μr = 100 (b) and a closed core(μr = 100) (c). Blue lines show a spherical discharge chamber with acylindrical hole.

4


We have simulated spatial distributions of electro-magnetic fields for this configuration by solving equation (8)for the vector magnetic potential with a source current fromthe primary coil inserted into a glass sphere of radius R. Thevector magnetic potential was set to zero at the boundaries ofthe computational domain, which was selected to be largeenough to ensure that the solution did not depend on theboundary position. We performed simulations of electro-magnetic fields without plasma for different lengths of thecoil. Indeed, we have observed that with increasing length ofthe solenoid from h = a (a current loop) to h > a (longsolenoid) the magnetic field became gradually confined insidethe solenoid, and leaked outside only through the coil ends.Most of the magnetic lines closed at a distance large com-pared to a. The amplitude of the magnetic field outside of thesolenoid gradually becomes much smaller than its valueinside.

To illustrate effects of ferromagnetic materials on spatialdistributions of the electromagnetic fields, we used differentmagnetic permittivity r of the core inside the tube. Figure 1shows the results of simulation for the coil length h = 3 cm,tube radius a = 1.25 cm and the glass sphere radiusR = 3.5 cm, which are typical to fluorescent light sources.Figure 1 compares spatial distributions of the magnetic linesand one contour line of A (red line) obtained for a 1 kA turncurrent (an Ampere-turn value) for different geometries of theferrite core. This red line indicates the position where A fallsto 1/e = 0.368 of its maximal value Am (at a point closeto r = a).

To understand the observed effects of the ferrite core onspatial distribution of magnetic fields it is useful to recall thatmagnetic fields near ferromagnetic materials behave similar toelectric fields near dielectrics. If an H (or E) field line enters amedium with a smaller magnetic or dielectric coefficient, it isrefracted towards the normal [28]. This is clearly seen infigure 1, which shows five equidistant magnetic lines selectedat the same radial interval inside the core. Magnetic inductionB is nearly uniform inside the middle of the solenoid (seefigure 2) and is higher by a factor of μeff inside the ferritecore. With increasing μr the magnetic field lines becomenormal to the ferrite boundary (see figure 1(b)). With closingthe core, they are fully confined inside the ferrite core(figure 1(c)).

Figure 2 shows the radial distributions of normalizedmagnetic induction B (figure 2(a)) and the vector magneticpotential A (figure 2(b)) in the mid-plane of the system for thethree cases shown in figure 1. It is seen in figure 2(a) that themagnetic field jumps at the coil position (at r = 1.25 cm) andincreases inside the closed core (at r ∼ 5 cm). The magneticfield inside the spherical chamber is small whereas the vectormagnetic potential and the corresponding induced electricfield (for AC case) are comparable to those inside the sole-noid. The maximal values of the B field in these three casesare 0.031, 0.349, and 1.517 T that corresponds to effectiveμeff of 11.26 and 50 for the last two cases. Insertion of aferromagnetic core of permeability μr inside the induction coil(with the same current) increases its inductance and inducedmagnetic flux in μeff times. For non-closed core magnetic

path, μeff = μr, and for μr ? 1, μeff practically does notdepend on μr, but is defined by the ratio of the core length/diameter.

3.2. Simulations of spherical ICP

For simulations of ICP produced inside the glass sphereshown in figure 1, we specified the coil current in the formj j tsin .0 ( ) The value of j0 was adjusted to obtain steadyplasma with a density of about 1012 cm−3 at 100 kHz. A fluidplasma model was used in simulations assuming MaxwellianEEDF, Joule electron heating, and the reaction rates for argongas given in the appendix. The fluid plasma model providedelectrical conductivity used by the electromagnetic solver.With air core, the RF current of 3.2 kA turn was required toobtain steady plasma (the initial plasma density eitherincreased or decreased for higher or lower currents). In thepresence of a ferromagnetic core, the coil current wasadjusted by a factor corresponding to the effective eff toobtain the same plasma density.

The calculated radial distributions of normalized electricfields in the ICP mid-plane are shown in figure 2(b) for argon

Figure 2. Radial distributions of normalized magnetic field B (a) andvector magnetic potential A (b) in the midplane of the system.Dashed lines show the induced electric field E in the AC case withplasma (see next section).

5


pressure p = 1 Torr and frequency f = 100 kHz. As seen infigure 2(b), the electric field distributions, E A t, arepractically the same as the radial distributions of the vectormagnetic potential in the three static cases shown by solidlines. Thus, the presence of plasma has a negligible effect onthe radial distributions of the fields. The calculated value ofthe skin depth at f = 100 kHz for plasma density 1012 cm−3 isδ ≈ 30 cm, which is much larger than plasma size, Λ ≈ 3 cm.This confirms the absence of a skin effect.

Figure 3 shows spatial distributions of the E field, theplasma density and the RF current density for the three cases.It is seen that the spatial distributions of E are quite different:the electric field is highly non-uniform in the case with aircore, and becomes more uniform with the introduction of aferromagnetic core and by closing the magnetic circuit. Theamplitude of the electric field required to maintain plasma inthe considered three cases are determined by integral char-acteristics of the electric field inside plasma, which are

determined by the geometry of the system and do not dependon plasma density.

3.3. Discussion and summary

Our simulations of electromagnetic fields produced by sole-noid of finite length have demonstrated that for a shortsolenoid, h a, the ratio of electric and magnetic fields isnearly uniform in space. The magnetic field dominates overelectric field at low frequencies, as E cB a c 1.∣ ( )∣For a long solenoid, h a, this ratio becomes non-uniform inspace due to the confinement of the B field inside thesolenoid.

For an ideal, infinite solenoid, analytical solutions can beobtained from the full Maxwell equations to illustrate effectsof displacement and conduction currents. The magneticpotential induced by azimuthal surface current, I t( )I tcos ,0 ( ) can be expressed in the frequency domain asA r t A r t, exp i ,( ) ( ) ( ) where A r( ) is given by theequation [22]:

rrrA

rk r A

d

d

d

d1 0. 92 2⎜ ⎟⎛

⎝⎞⎠ ( ) ( )

Here k2 2 and ( ) is the complex permittivity ofmedia. The general solution of this differential equation fordistributions of electric and magnetic fields expressed in termsof Bessel functions are discussed in [22, 29] for the vacuumcase, when c, , and .0 0 0 0

2

Let us consider the near-field and far-field regions whereequation (9) can be simplified:

rrrA

rA k r

rrrA

rk r A k r

d

d

d

d0, 1

d

d

d

d0, 1. 102 2

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

∣ ∣

∣ ∣ ( )

The solutions of equation (10), which are applicable forcomplex values of the wave vector k are:

A r

B rr a

B a

ra r k

C kr

rr k

2,

2, 1

exp i, 1

, 11

0

02

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

( ) ∣ ∣

( ) ∣ ∣

( )

where B I N ,0 0 0 and C is a constant. The magnetic andelectric fields are expressed as:

Br r

rA

E A

1 d

d,

i . 12

( )

( )The far-field solution corresponds to a cylindrical wave invacuum or to an exponentially damped cylindrical wave inplasma. For a cold plasma, at , the wave vectork ci pe is purely imaginary, and the classical skin depth is

c ,pe where pe is the electron plasma frequency. In thecase of the cylindrical wave in vacuum, E cB 1,∣ ( )∣whereas in the plasma case, E cB 1.pe∣ ( )∣ In theplasma case, the magnetic field generated by plasma within

Figure 3. Spatial distributions of the electric field (contours), plasmadensity (on the right) and AC current density (on the left) for the aircore (top), a ferrite core of finite length (center) and a closed ferritecore (bottom). Gray areas show regions with μr = 100.

6


the skin layer exceeds the magnetic field produced by theprimary current.

In the near-field region, the magnetic and electric fieldsare given by [22]:

B Bt r a

a r kcos ,0, 1

130{ ∣ ∣ ( )

E B

rt r a

a

rt a r k

2sin ,

2sin , 1

. 140 2

⎛

⎝

⎜⎜⎜⎜ ∣ ∣( )

Therefore, the magnetic field inside the infinite solenoiddominates over the electric field, as E cB r c 1,∣ ( )∣whereas the electric field becomes dominant outside ata r k1 , where the B field is zero. For typical condi-tions, a ∼ 1 cm, f 10 104 6 Hz, the ratioa c 2 10 106 3( ) is a very small number. Thus, for

all practical purposes, one can neglect the magnetic fieldoutside of long solenoids. If plasma is located outside thesolenoid, the magnetic field in plasma can be neglected.Contrary, if plasma is located inside the solenoid, magneticfield in plasma may be substantial, and the plasma dynamicscould be nonlinear at low driving frequencies.

Insertion a ferromagnetic core with open ends inside thesolenoid leads to increasing the B field inside the core, andfurther reduction of the magnetic field outside the core(figure 2(a)). Closing the magnetic circuit leads to betterconfinement of the magnetic field within the core and itscomplete absence in plasma. In the last case, only themagnetic field created by the plasma current is present outsidethe core. This magnetic field is negligible for the plasmadensities considered in our cases. Exactly such situationoccurs in a conventional low-frequency transformer with highpermeability core 10 .r

4( )The skin effect is absent al low driving frequencies. The

collisional skin depth at a frequency of 100 kHz and plasmadensity 1012 cm−3 considerably exceeds the plasma size. As aresult, plasma has very little effect on spatial distributions ofthe electric and magnetic fields (compare radial distributionsof the fields in figure 2 with and without plasma).

The alternating plasma current induces a magnetic fluxΦp outside the plasma that co-interacts with the externalmagnetic flux produced by the primary coil, Φc. The resultantfluxΦ =Φc–Φp provides the plasma electric field required bythe ionization and energy balances in plasma. Since theresultant magnetic flux Φ is coupled to the antenna coil, themagnetic flux induced by the plasma current reduces themagnetic flux in the inductor, thus affecting its impedance,mainly in raising its resistance.

A softer dependence n I0( ) takes place in high-frequencyICP, when δ < Λ. The stabilizing effect is due to the skineffect slowing the increase of plasma density with increasingcurrent. Generally, driving low frequency ICP with a currentsource is unstable since I n n Id d 10 0( )( ) due to the lack ofshielding effect, and raising inductor resistance caused byplasma loading.

The spatial distributions of the electric and magneticfields are highly non-uniform and are determined by theprimary coil configuration and by the presence of ferromag-netic materials. In experiments, the amplitude of the electricfield required for plasma maintenance satisfying ionizationand electron energy balances is selected via coupling of pri-mary and secondary currents. In our simulations, the coilcurrent (Ampere-turn value) was adjusted according to μeff toprovide the electric fields required by plasma in such a waythat the plasma density remained nearly constant over a longtime compared to the rf period. In future work, the couplingmechanism at low frequencies, in the absence of the skineffect, can be implemented by specifying the period averagedpower absorbed in plasma and adjusting the coil current tosustain this power.

4. Simulation of ICP dynamics

The second ICP topology we considered in this paper corre-sponds to FMICP with ring ferrite cores shown in figure 4[17, 18]. Due to toroidal ferrites with large permeability(μr = 2000) and closed magnetic path, the magnetic fieldoutside the core is about μr times smaller than the magneticfield inside the core producing EMF in plasma. That makesthis FMICP working as an ideal transformer in spite of theferromagnetic cores embrace only a small part of the plasmacurrent path. Therefore, the discharge current I0 = 2IcN,where Ic is the current of the N-turn winding of each toroidalcore. In this respect, this FMICP is equivalent to an ACdriven plasma column without an external AC magnetic field.

In our simulations, we did not solve the Maxwellequations. Instead, we specified the discharge current, andcalculated the alternating electric field from Ohm’s law:

E tI t

b n r t r r

sin

2 , d. 15

R

e e

0

0

( ) ( )

( )( )

Here I0 is the amplitude of the discharge current, and be is theDC value of electron mobility.

We simulated radial distributions of plasma species andelectron temperature in a tube of radius R = 1 cm using a fluidmodel at different driving frequencies and different gaspressures. The gas phase chemical reactions used in oursimulations for argon gas are shown in the appendix. Therates of electron induced reactions were calculated based oncollision cross-sections and either assuming a Maxwellian

Figure 4. MFICPs with closed magnetic path as light source.

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EEDF or solving (local) Boltzmann equation for electrons(see below).

Below, we show the time evolution of the electric field,electron temperature and plasma density on the discharge axisfor a periodic state established after a few periods. Figure 5shows simulation results at pressure of 2 Torr and I0 = 0.1 Afor different angular frequencies ω. The characteristic fre-quencies are ν = 2 × 109 s−1, νa = 4 × 103 s−1 and νε = 2 ×105 s−1 for the time averaged plasma density 3.0 × 1011 cm−3

and electron temperature 2 eV.The lowest frequency ω = 100 s−1 (ω/νa ≈ 0.03) cor-

responds to a quasi-static regime. In this regime, the electricfield remains nearly constant during the current half-period,while the plasma density changes in phase with the dischargecurrent (see figure 5). The electron temperature is nearlyconstant during the half-period, and drops sharply near cur-rent zero when the electric field changes sign. With anexception of the short time interval near current zero, theelectric field and plasma parameters behave similarly to thosein a DC positive column with a slow current variation. Fromthe ionization and electron energy balance of DC positivecolumn we know that electron temperature is defined by thepΛ product, and depends weakly on the current, while theplasma density is nearly proportional to the discharge current.The proportionality of plasma density and plasma con-ductivity to discharge current is the primary reason for thenearly constant electric field in the quasi-static regime of ICPoperation at ω = νa.

With increasing frequency, ICP is entering into adynamic regime. In this regime, the modulation of the plasmadensity decreases when ω becomes comparable to νa, andgradually disappears at ω? νa. At ω = 104 s−1 when ω/νa ≈3, the plasma density modulation is about 30%, and for ω =105 s−1 and higher (ω/νa � 30) the plasma density modula-tion disappears completely. A different behavior is observedfor the electron temperature in the dynamic regime(figure 5(b)). The modulation of Te increases with increasingfrequency, reaching its maximum at a frequency ω close tothe electron energy loss frequency, νε, and disappears withfurther frequency increases. In figure 5(b) the maximalmodulation of Te is observed at ω = 105 s−1 when ω/νε ≈ 2.

The high-frequency regime corresponds to negligiblemodulations of both the plasma density and electron temp-erature. This regime is typical to industrial ICPs withdriving frequency ω > 107 s−1, which corresponds to f =ω/2π 2 MHz.

Results for a considerably lower argon pressure,p = 0.1 Torr are shown in figure 6. The time evolution ofplasma parameters is similar to the previous case, but shiftedto higher frequencies ω due to increased characteristic fre-quencies νa and νε. This increase is caused by raising theelectron temperature at lower gas pressures in spite ofdecreasing electron collision frequency, ν. It is seen infigures 5 and 6 that the electric field and electron temperatureovershoots (more pronounced at lower gas pressure) arecaused by the delay in plasma density build up after thecurrent reversals.

Figure 7 summarizes the pressure dependence for the samedriving frequency ω = 105 s−1, ( f = 16 kHz). Simulations for0.01 and 0.1 Torr were performed by solving the full ionmomentum equation, whereas simulations for gas pressures of1 Torr and above used the drift-diffusion approximation forions. For the same current, I0 = 0.1 A, the electric field andelectron temperature increase with decreasing gas pressure, andplasma density decreases with decreasing pressure; a verycommon trend in any gas discharge plasma dominated by thecharge particle loss at the wall. As seen in figure 7, for different

Figure 5. Time dependence of the electric field (a), electrontemperature (b), and plasma density (c) on axis of the tube fordifferent frequencies at 2 Torr and 0.1 A. For clarity, only electrontemperature is shown for ω = 107 s−1.

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gas pressures, all three ICP regimes (quasi-static, dynamic andhigh-frequency) can be realized for the same driving frequency.

5. Comparison with experiments and discussion

Experimental studies of low frequency ICPs, especially theirdynamic characteristics, have been performed only over alimited range of conditions. This section will offer somequalitative comparisons of simulation results with available

experimental observations and discuss trends, rather thanproviding a detailed quantitative picture.

5.1. Experimental studies of low frequency ICP

Time resolved electron energy probability functions (EEPFs)measurement (with resolution of 5 × 10−7 s) were performedin the electrodeless lamp (shown in figure 4) filled with amixture of Kr (300 mTorr) and Hg (6 mTorr), and driven at50 KHz in the tube with radius R = 2.5 cm and a closeddischarge path L = 80 cm [30].

Figure 6. Time dependence of the electric field (a), electrontemperature (b), and plasma density (c) on axis of the tube fordifferent frequencies at 0.1 Torr and 0.1 A.

Figure 7. Time dependence of the electric field (a), electrontemperature (b), and plasma density (c) on axis for differentpressures at frequency 16 kHz.

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Time dependent electromotive force (actually, the mod-ule of the lamp voltage), |VL|(t), and electron temperature,Te(t), found as an integral of the measured EEDF at I0 = 2; 4and 8 A are presented in figure 8. The EEPFs measured at themoments of minimal and maximal electron temperature, areshown in figure 9.

A simple model was used in [29] to calculate the timevarying electron temperature from the equation:

T

t

e E t

mT T . 16e ce e

2 2 ( ) ( ) ( )

The frequency of electron energy loss Tce( ) was assumed to

be a constant corresponding to the averaged over AC periodvalue of Te (dashed lines in figure 10), and the rms value of E.From this equation, it becomes clear that the electron temp-erature follows the electric field during the main part of thehalf-period and relaxes with a time scale c 1( ) near currentzero, when e E t T m.c

e2 2 ( ) Fast electron heating, and

rapid electron cooling occur at the times of low dischargecurrent is seen in figure 10 and more pronounced in figures 6–8.An asymmetry in the heating and cooling rates is due to thelarger electric fields at lower discharge currents (the well-known negative differential V/A characteristic of dischargeplasmas) and the field-independent electron cooling near zerocurrent. Although the results of simple modeling shown infigure 10 are in plausible agreement with the experiment, theydo not show the highly pronounced time asymmetry in Te(t)waveform as in figures 6–8 obtained with more accurate model.

5.2. Effects of non-Maxwellian EEDF

Due to similarities between the positive column of DC dis-charges and FMICP at low frequencies, it is expected thatseveral phenomena previously studied in DC dischargesshould be observed in the FMICP. In particular, under certainconditions, plasma constriction and stratification are expectedto occur in the FMICP. In fact, plasma stratification in the

induction neon lamp reported in figure 51 of [19] correspondsto the conditions of positive column stratification in DCdischarges at similar values of pR and I0 [31]. Indeed,standing striations have been observed in ICP in contrast tothe moving striations in DC discharges because the DCcomponent of the electric field responsible for the striationmotion is absent in ICP.

To illustrate some of the anticipated kinetic effects, wehave performed simulations of the ICP column for non-Maxwellian EEDFs using look-up-table technique [32]. Thelocal Boltzmann solver for electrons was used to simulate theEEDF and compute the electron transport coefficients andrates of electron-induced chemical reactions over a range ofelectric fields (electron temperature) and plasma densities.

s

Figure 8. Time dependence of discharge voltage and electrontemperature during rf period. The numbers show discharge currents.

Figure 9. EEPF measured at the moment of minimal (t = 1 μs) andmaximal. (t = 5 μs) electron temperature.

Figure 10. Electron temperature calculated according toequation (16) (solid lines) and those found from the measured EEDFin Ar/Hg mixture [30].

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The later were used instead of the rates calculated with aMaxwellian EEDF.

Figure 11 shows the typical radial distributions of theplasma density in ICP at 16 KHz for argon gas pressures of 1and 10 Torr and discharge currents 0.1 and 1 A. The calcu-lated values of plasma density on the axis for these conditionswere n0 = 2.37 × 1011 cm−3 for 1 Torr, 0.1 A, n0 = 3.53 ×1011 cm−3 for 10 Torr, 0.1 A, and n0 = 4.62 × 1013 cm−3 for10 Torr, 1 A. For the low current, low-pressure case, the radialdistribution of plasma density resembles the well-knownSchottky solution. At low current, a broadening of radialdistribution takes place with increasing pressure, whenvolume recombination of molecular ions comes into play.With increasing current, a constriction of plasma towards theaxis takes place at 10 Torr due to Maxwellization of theEEDF. Such a behavior is typical for positive column plasmaof DC discharges in noble gases [33].

6. Conclusions

Although ICP has been studied for more than a century, plasmadynamics at low driving frequencies remain poorly understood.In particular, the role of strong alternating magnetic fields at lowfrequencies is completely unexplored. We have pointed out thatdepending on ICP topology, the B fields could be negligiblysmall in plasma (as in the FMICP with closed ferromagnetic ringcore) or play a major role in plasma dynamics (as in the sphe-rical theta pinch). We have considered the first case in somedetail and left the second case for future studies.

We have clarified an apparently paradoxical situationthat, depending on primary coil configuration and the pre-sence of ferromagnetic materials, the ratio of E and B fields,E cB∣ ( )∣ can be highly non-uniform in space. For some ICPconfigurations, the B field may be negligibly small in plasma,whereas the E field is sufficient for plasma generation. Cal-culations confirmed that at low frequencies, the presence ofplasma has very little effect on spatial distributions of theelectric and magnetic fields due to the absence of the skin

effect. Under these conditions, the field distributions aredetermined solely by the coil geometry and by the presence offerromagnetic materials. However, the amplitude of theelectric field in plasma is controlled by the plasma itself,which provides a transformer effect via coupled magneticfluxes. In our simulations, the coil current (Ampere-turnvalue) was adjusted to find the electric fields required by theplasma equilibrium in such a way that plasma densityremained nearly constant in simulations over a long timecompared to the RF period. In future work, we plan toimplement capabilities to automatically adjust the coil currentfor a given power absorbed in plasma over the field period.

To illustrate plasma dynamics in quasi-static, dynamicand high-frequency regimes, we have performed 1D simula-tion of an ICP transformer over a wide range of driving fre-quencies and gas pressures. For this ICP configuration havinga ferromagnetic core with a closed magnetic circuit, thealternating magnetic field created by the primary coil iscompletely absent in plasma. The magnetic field created bythe plasma current is mainly confined within the core, pro-viding a feedback for ICP stabilization. While the fluid modelwas capable of illustrating different plasma dynamics in thethree regimes, and to explain trends observed in the experi-ments, quantitative comparison may require kinetic treatmentof electrons. This may be particularly important for thecollisionless electron heating and nonlinear electrodynamicsat low pressures, for fast electron cooling near current zero,etc. As an example of important kinetic effects, we haveillustrated in our simulations that for non-Maxwellian EEDF,dynamic constriction of ICP column can be observed.

Further development of multi-dimensional ICP modelsare required to understand configurations (such as ICP withplanar coils) where alternating B and E fields in plasma couldresult in complicated plasma dynamics at low frequencies.

Acknowledgments

This work was supported by the Plasma Science Centeroperating under the US Department of Energy Office ofFusion Energy Science Contract DE-SC0001939. VIK thanksDr Robert Arslanbekov for help with simulations and OlivierMorisot of ESI group for providing a trial version of the latestCFD-ACE+ software. We wish to thank the referees and theEditor for useful comments, which helped improve the paper.

Appendix. Gas phase chemical reactions used insimulations

Gas-phase chemical reactions used in simulations of argonplasma are shown in the table below. The rates of electron-induced reactions (1)–(10) were calculated using corresp-onding collision cross-sections with a Maxwellian EEDF orsolving (local) Boltzmann equation for the EEDF.

Figure 11. Radial distributions of plasma density in ICP column at16 kHz for two gas pressures and currents.

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ORCID

Vladimir I Kolobov https://orcid.org/0000-0002-6197-3258

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Inductively coupled plasmas at low driving frequencies - DOE …doeplasma.eecs.umich.edu/files/PSC_Kolobov6.pdf · 2018. 2. 2. · Inductively coupled plasmas at low driving frequencies