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Symmetry breaking in high frequency, symmetric capacitively coupled plasmas E. Kawamura, M. A. Lieberman, and A. J. Lichtenberg Citation: Physics of Plasmas 25, 093517 (2018); doi: 10.1063/1.5048947 View online: https://doi.org/10.1063/1.5048947 View Table of Contents: http://aip.scitation.org/toc/php/25/9 Published by the American Institute of Physics

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  • Symmetry breaking in high frequency, symmetric capacitively coupled plasmasE. Kawamura, M. A. Lieberman, and A. J. Lichtenberg

    Citation: Physics of Plasmas 25, 093517 (2018); doi: 10.1063/1.5048947View online: https://doi.org/10.1063/1.5048947View Table of Contents: http://aip.scitation.org/toc/php/25/9Published by the American Institute of Physics

    http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1914574873/x01/AIP-PT/COMSOL_PoPArticleDL_WP_0818/comsol_JAD.JPG/434f71374e315a556e61414141774c75?xhttp://aip.scitation.org/author/Kawamura%2C+Ehttp://aip.scitation.org/author/Lieberman%2C+M+Ahttp://aip.scitation.org/author/Lichtenberg%2C+A+J/loi/phphttps://doi.org/10.1063/1.5048947http://aip.scitation.org/toc/php/25/9http://aip.scitation.org/publisher/

  • Symmetry breaking in high frequency, symmetric capacitively coupledplasmas

    E. Kawamura,a) M. A. Lieberman, and A. J. LichtenbergDepartment of Electrical Engineering and Computer Sciences, University of California, Berkeley,California 94720, USA

    (Received 18 July 2018; accepted 6 September 2018; published online 24 September 2018)

    Two radially propagating surface wave modes, “symmetric,” in which the upper and lower axial

    sheath fields (Ez) are aligned, and “anti-symmetric,” in which they are opposed, can exist in capaci-tively coupled plasma (CCP) discharges. For a symmetric (equal electrode areas) CCP driven sym-

    metrically, we expected to observe only the symmetric mode. Instead, we find that when the

    applied rf frequency f is above or near an anti-symmetric spatial resonance, both modes can exist incombination and lead to unexpected non-symmetric equilibria. We use a fast 2D axisymmetric

    fluid-analytical code to study a symmetric CCP reactor at low pressure (7.5 mTorr argon) and low

    density (�3� 1015 m�3) in the frequency range of f¼ 55 to 100 MHz which encompasses the firstanti-symmetric spatial resonance frequency fa but is far below the first symmetric spatial resonancefs. For lower frequencies such that f is well below fa, the symmetric CCP is in a stable symmetricequilibrium, as expected, but at higher frequencies such that f is near or greater than fa, a non-symmetric equilibrium appears which may be stable or unstable. We develop a nonlinear lumped

    circuit model of the symmetric CCP to better understand these unexpected results, indicating that

    the proximity to the anti-symmetric spatial resonance allows self-exciting of the anti-symmetric

    mode even in a symmetric system. The circuit model results agree well with the fluid simulations.

    A linear stability analysis of the symmetric equilibrium describes a transition with increasing fre-

    quency from stable to unstable. Published by AIP Publishing. https://doi.org/10.1063/1.5048947

    I. INTRODUCTION

    High frequency, low pressure, axisymmetric capaci-

    tively coupled plasma (CCP) reactors are widely used in the

    semiconductor processing industry but can exhibit electro-

    magnetic (EM) effects which limit processing uniformity.1–5

    Lower pressures reduce collisions, leading to an improved

    ion anisotropy at the wafer target. Higher frequencies result

    in decreasing the sheath widths and voltages, leading to a

    decrease in the ion bombarding energy, which may be desir-

    able for processing integrated circuits with smaller dimen-

    sions. Numerical simulations which solve Maxwell’s

    equations in the frequency domain self-consistently with the

    plasma transport in two dimensions have been used to study

    EM effects in high frequency discharges.6–11 Results have

    also been obtained from more sophisticated simulations

    which solve Maxwell’s equations in the time-domain and

    capture non-linear effects.12–16 The main conclusion is that

    at higher frequencies and/or larger areas, the wavelengths of

    the EM surface waves in the plasma can become shorter than

    the reactor radius, leading to standing wave effects and con-

    sequent plasma non-uniformities. For intermediate frequen-

    cies, above the typical drive frequency of 13.56 MHz but

    well below the first spatial resonances of the waves, non-

    linearly generated harmonics can also become resonant or

    near-resonant.10,17–20

    The electrode/plasma/electrode sandwich structure of a

    symmetric (equal electrode areas) cylindrical discharge

    forms a three electrode system in which both z-symmetric

    and z-anti-symmetric radially propagating wave modesexist.5,9,21–28 The upper and lower axial sheath fields (Ez) arealigned for the symmetric mode, while they are opposed for

    the anti-symmetric mode. In Ref. 9, a linear analytic EM

    model was used to study the modes of a non-symmetric

    cylindrical CCP reactor with uniform electron density n andunequal uniform sheath widths sb and st at the bottom andtop electrodes, respectively. The radial wavenumbers of the

    symmetric and anti-symmetric modes were found to be9

    ks �xc

    L

    sb þ st

    � �1=2(1)

    and

    ka �xxp

    sb þ stsbstD

    � �1=2; (2)

    respectively, with c being the speed of light, x being theapplied radian frequency, L being the discharge width, Dbeing the bulk plasma width, xp ¼ ðne2=ð�0mÞÞ1=2 beingthe electron plasma frequency, e being the elementarycharge, �0 being the free space permittivity, and m being theelectron mass. At higher frequencies and lower sheath

    widths, both ks and ka are larger, i.e., the correspondingwavelengths ks ¼ 2p=ks and ka ¼ 2p=ka are smaller,increasing wave effects. The first symmetric mode spatial

    resonance occurs at ks R¼ v01 with R being the dischargeradius and v01¼ 2.405 being the first zero of the zerothorder Bessel function, while the first anti-symmetric mode

    spatial resonance occurs at kaR¼ v11 with v11 � 3:832a)[email protected]

    1070-664X/2018/25(9)/093517/13/$30.00 Published by AIP Publishing.25, 093517-1

    PHYSICS OF PLASMAS 25, 093517 (2018)

    https://doi.org/10.1063/1.5048947https://doi.org/10.1063/1.5048947https://doi.org/10.1063/1.5048947mailto:[email protected]://crossmark.crossref.org/dialog/?doi=10.1063/1.5048947&domain=pdf&date_stamp=2018-09-24

  • being the first zero of the first order Bessel function. Thus,

    the first resonance frequencies for the symmetric and anti-

    symmetric modes are given by

    fs �v01c2pR

    sb þ stL

    � �1=2(3)

    and

    fa � xpv112pR

    sbstD

    sb þ st

    � �1=2; (4)

    respectively. Since fa / xp, fa can be much smaller than fs atlow electron density.

    Since typical CCP reactors used in industry are non-

    symmetric with the powered wafer electrode significantly

    smaller than the grounded electrode, most recent works have

    studied non-symmetric discharges. Here, we use a fast 2D

    axisymmetric fluid-analytical code8,29 to study a low pres-

    sure, low density symmetric parallel-plate argon dischargewith parameters similar to a recent experiment.30 As in our

    simulations, the experiment observed standing wave effects,

    leading to center-high plasma non-uniformities. However,

    symmetry breaking could not be explored in this experiment,

    as its reactor geometry had significant asymmetry. We exam-

    ine the discharge at a low pressure of 7.5 mTorr to avoid

    wave damping phenomena due to collisions. The discharge

    is driven at low powers by a single high frequency source in

    the range of f¼ 55–100 MHz, keeping the average electrondensity n� 3� 1015 m�3. At this low density, fa � fs, andthe above frequency range encompasses the first anti-

    symmetric mode resonance frequency fa, but is far below thefirst symmetric mode resonance frequency fs. At lower f sig-nificantly below fa, the symmetric discharge is in a stablesymmetric equilibrium, as expected. However, driving the

    discharge at higher f, near or above fa, can lead to unex-pected non-symmetric equilibria even for symmetrically

    driven systems. Reference 6 showed 2D fluid simulation

    results of a symmetric CCP reactor operated at low pressure

    (2 mTorr) and high frequency (80 MHz). However, non-

    symmetric equilibria were not observed in that reactor since

    it was operated at a much higher power (900 W) and density

    (�1� 1017 m�3), so that fa � fs was not satisfied. Due toour much lower operating power and density, we did not

    observe the “skin effect” described in this reference. Also,

    this and other references describe the electron heating due to

    the axial (Ez) and radial (Er) fields as “capacitive” and“inductive,” respectively.

    This paper is organized as follows: in Sec. II A, we

    briefly describe the self-consistent 2D axisymmetric fluid-

    analytical simulation of a symmetric parallel-plate CCP

    reactor. In Sec. II B, we present and discuss the simulation

    results in the range f¼ 55–100 MHz, which includes fa butis well below fs. In Sec. III A, we develop a lumped circuitmodel of the symmetric CCP, and in Sec. III B we compare

    the model results with the fluid simulations. In Sec. III C,

    we simplify the circuit model in order to analyze the stabil-

    ity of the symmetric equilibria. Conclusions are given in

    Sec. IV.

    II. SYMMETRIC CCP DISCHARGE SIMULATIONS

    A. 2D fluid-analytical simulation description

    The fast 2D fluid-analytical CCP simulation has been

    described in detail previously,8,29 so we only give a brief sum-

    mary below. The simulation was developed using the finite ele-

    ments tool COMSOL in the Matlab numerical computing

    environment. The CCP configuration for the symmetric dis-

    charge studied is shown in Fig. 1. We assume an axisymmetric

    cylindrical geometry with center of symmetry at r¼ 0 (z-axis).The bulk plasma region of width D¼ 2.4 cm is surrounded by asheath region with a nominal width of s0¼ 3 mm. The electrodespacing and radius are L¼Dþ 2s0¼ 3 cm and R¼ 10 cm,respectively. We use the argon cross section set compiled by

    Vahedi and Surendra31 to calculate the reaction rate coefficients

    assuming a Maxwellian electron energy distribution.

    The simulation treats each region of the reactor as a

    dielectric slab. The free-space magnetic permeability l¼l0 isassumed everywhere, while �¼ j�0 depends on the relativedielectric constant j of each region. The sheath relativedielectric constant js is initially set to one but is calculated asa function of the local electric field and other plasma parame-

    ters, to keep s¼ s0, a constant, as discussed in Refs. 8 and 29.In the plasma region, the relative dielectric constant is

    jp ¼ 1�x2p

    xðx� j�mÞ; (5)

    where x¼ 2pf is the applied radian rf frequency and �m isthe electron-neutral momentum transfer collision frequency.

    Note that jp is complex with a dissipative imaginary compo-nent, so the plasma region is a lossy dielectric.

    In axisymmetric geometry, the capacitive fields Er, Ez,and H/ are in the transverse magnetic (TM) mode. In thiscase, the magnetic field is transverse to the axis of symmetry,

    while the electric field has components both parallel and

    transverse to the axis of symmetry. All the field components

    are proportional to ejxt. This eliminates the time-dependence

    from the field solve so that the time-independent Helmholtz

    equation can be used to solve for the fields, simplifying and

    speeding up the EM simulations, but at the cost of ignoring

    any nonlinearly generated harmonics.

    The CCP is powered by applying an rf current of magni-

    tude Irf across the electrodes. To determine the capacitive fieldsresulting from the applied current Irf, we solve the Helmholtzequation in the entire domain, using the following boundary

    conditions on the dependent variable Iðr; zÞ ¼ 2prH/:

    FIG. 1. Geometry of the symmetrically driven capacitive discharge used in

    the 2D fluid-analytical simulations.

    093517-2 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • I ¼ 0 on center of symmetry; (6)

    n̂ � rI ¼ 0 on all conducting walls; (7)

    I ¼ Irf ¼ const at r ¼ R: (8)

    Since I / r by definition, I must be zero at the center of sym-metry. The second condition is equivalent to setting the tan-

    gential electric field at the conducting walls to zero. From

    Ampere’s law, I(r) gives the total current flowing normal tothe cross-sectional area enclosed by a loop of radius r. Thus,the third condition sets the total applied current in the dis-

    charge to Irf.For these simulations, we hold the power Pe absorbed

    by the electrons to a fixed value, in order to keep the average

    electron density n at a fixed value. To simulate a system witha fixed Pe¼Pe0, we solve for the EM fields and then calcu-late Pe. If Pe is not equal to Pe0, then we adjust Irf and repeatthe EM solve until Pe¼Pe0 within a previously set relativetolerance level. The CCP simulations consisted of three basic

    parts: (1) a linear EM calculation which uses the time-

    independent Helmholtz equation to solve for the capacitive

    fields in the linearized frequency domain; (2) an ambipolar,

    quasineutral bulk plasma calculation which solves the time-

    dependent fluid equations for ion continuity and electron

    energy balance; and (3) an analytical sheath calculation

    which solves for the sheath parameters (i.e., sheath voltage,

    sheath width, and js). The total simulation time for the reac-tor is about 20 min on a moderate workstation with 2.7 GHz

    central processing unit (CPU) and 12 GB of memory.

    B. 2D fluid-analytical simulation results

    We present and discuss the simulation results for a 7.5

    mTorr argon CCP reactor shown in Fig. 1 over the frequency

    range of f¼ 55 to 100 MHz in 5 MHz intervals. The simula-tions assume the collisionless Child Law type sheath model

    derived in Ref. 32, since it provides a self-consistent solution

    for a capacitive rf sheath in this low pressure, high frequency

    operating regime. The current source magnitude Irf isadjusted to keep Pe � 5 W, resulting in an average electrondensity of about n� 3� 1015 m�3. The discharge is drivensymmetrically, and the simulation is started with symmetric

    initial conditions about the midplane at z¼ 1.5 cm. Forf¼ 55, 60, and 65 MHz, the discharge reached a symmetricalsteady state about the midplane. At f¼ 70 and 75 MHz, thedischarge did not reach a stable equilibrium but oscillated

    between symmetric and non-symmetric states. At f¼ 80, 85,90, and 95 MHz, the discharge reached a non-symmetric

    steady state, which was surprising for a symmetrically driven

    system starting with symmetric initial conditions. At

    f¼ 100 MHz, the discharge did not reach a steady state andoscillated between symmetric and non-symmetric states, as

    in the f¼ 70 and 75 MHz cases.Figure 2 shows the fluid simulation results versus r at

    f¼ 60 MHz for (a) the electron density n at the midplane (dot-ted) and along the bottom (solid) and top (dashed) sheath

    edges, (b) the electron temperature Te along the bottom

    (solid) and top (dashed) sheath edges, (c) the rf sheath voltage

    amplitudes Vsh, and (d) the time-averaged sheath widths s at

    the bottom (solid) and top (dashed) electrodes. As expected

    for a symmetrically driven CCP with symmetric initial con-

    ditions, all the diagnostics are symmetric about the mid-

    plane at z¼ 1.5 cm, so that their values at the bottom (solid)and top (dashed) electrodes overlap. The sheath voltages

    are fairly uniform except for spiking near the corners of the

    discharge at the electrode edges (electrostatic edge effect6).

    The sheath width variation closely follows that of the sheath

    voltage since the analytical sheath calculation assumes a

    Child Law relation s / V3=4sh . Except for the density profilewhich is due to radial diffusion, these diagnostics do not

    display any significant radial variations, indicating that the

    discharge frequency is below both its spatial resonances.

    The simulation results (not shown) at f¼ 55 and 65 MHzare similar to those shown in Fig. 2. From (4) and (3), for

    f¼ 55–65 MHz, fa � 85–81 MHz and fs� 420–400 MHz.So, f < fa � fs for these cases which reached symmetricsteady states.

    Figure 3 shows the fluid simulation results versus r atf¼ 80 MHz for the same diagnostics as shown in Fig. 2. Inthis case, the sheath parameters Vsh and s are non-symmetricabout the midplane and show significant radial variations. In

    contrast, the bulk plasma parameters n and Te are mostlysymmetric about the midplane due to the high discharge dif-

    fusivity at the low operating pressure of 7.5 mTorr. We

    note that the radius Ri � ðv01=v11ÞR � 6:3 cm represents atransition point for the sheath parameters, such that for r Ri; Vshb Vsht and sb st, while for r > Ri; Vshb > Vshtand sb > st. Here, we use the subscripts b and t to represent“bottom” and “top,” respectively. From (4) and (3), for

    f¼ 80–95 MHz, fa � 65–71 MHz and fs � 340–350 MHz,respectively. So, fa < f � fs for the simulated cases whichreached non-symmetric steady states. Since the gap between

    f and fa increases with increasing f, we expect the non-symmetry in the diagnostics to decrease with increasing f.Fluid simulation results (not shown) at f¼ 85, 90, and95 MHz are similar to those shown in Fig. 3, but with the

    non-symmetry decreasing with increasing f.Figure 4 shows the contour plots at (a) 60 MHz and (b)

    80 MHz of Iðr; zÞ ¼ 2prH/, which give the total currentflowing normal to the cross-sectional area enclosed by a loop

    of radius r. At f¼ 60 MHz, I(r, z) is symmetric about themidplane, and except for a small fringing effect near the

    radial edges, all the current is in the axial (z) direction so thatEr � Ez. Thus, for the symmetric steady states, the electronpower due to the radial fields Pr is negligible compared tothe total electron power Pe. For example, for the f¼ 55, 60,and 65 MHz cases, where the discharge reached a symmetric

    steady state, Pr=Pe � 0:003. At f¼ 80 MHz, I(r, z) is non-symmetric about the midplane, and there are significant

    radial fields and currents. In this case, Pr is non-negligible,and we found Pr=Pe � 0:38 for this case. For the other non-symmetric steady-state cases at f¼ 85, 90, and 95 MHz, wefound that Pr/Pe¼ 0.24, 0.13, and 0.032, respectively, con-firming that the non-symmetry decreased with increasing fre-

    quency away from fa.The axial fields Ez at the bottom and top electrodes are

    aligned for the symmetric mode, while they are opposed for

    the anti-symmetric mode. We define Ezs and Eza as the axial

    093517-3 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • sheath fields for the symmetric and anti-symmetric modes,

    respectively, while we define Ezb and Ezt as the sheath fieldsat the bottom and top electrodes, respectively. Then, Ezb¼ Ezs þ Eza and Ezt¼Ezs – Eza, giving

    Ezs ¼Ezb þ Ezt

    2(9)

    and

    Eza ¼Ezb � Ezt

    2: (10)

    In Fig. 5, we show the results versus r at (a) 60 MHz, (b)65 MHz, (c) 80 MHz, (d) 85 MHz, (e) 90 MHz, and (f)

    95 MHz for Ezs (solid) and Eza (dashed) at four differentphases / ¼ xt ¼ p=4; p=2; 3p=4, and p of an rf half-cycle.(The second half-cycle results are reflections across the

    Ez¼ 0 axis of the first half-cycle results.) For the frequenciesf¼ 55 (not shown), 60, and 65 MHz, in which the dischargereaches a symmetric steady state, Eza � 0 and Ezs(r) is fairlyuniform, indicating that the symmetric mode dominates and

    f � fs. For the frequencies f¼ 80, 85, 90, and 95 MHz, inwhich the discharge reaches a non-symmetric steady state,

    the axial sheath field has both symmetric and anti-symmetric

    mode components. For these cases, Eza shows significantradial variations, while Ezs is fairly uniform, indicating thatf � fa and f � fs. The Eza amplitude for the discharge is

    highest at f¼ 80 MHz which is nearest to its anti-symmetricmode resonance frequency fa, and then decreases withincreasing frequency. The proximity of the discharge to its

    first anti-symmetric mode resonance probably accounts for

    its instability at f¼ 70 MHz. At f¼ 100 MHz, the dischargeis also unstable and unable to reach a steady state, which

    may be due to the discharge approaching its second anti-

    symmetric mode resonance. These unstable cases will be dis-

    cussed further below. For the cases with non-symmetric

    steady states, Eza passes through zero and changes sign atr ¼ Ri � ðv01=v11ÞR � 6:3 cm, in agreement with Fig. 3 thatr¼Ri is a transition point for the discharge parameters.

    We also performed fluid simulations at Pe¼ 10 W whichis twice the electron power of the fluid simulations discussed

    above. The electron densities were correspondingly about

    twice as high, and fa from the scaling in (4) was aboutffiffiffi2p

    higher. We found similar phenomena to that shown in Fig. 5

    for Pe¼ 5 W but shifted to correspondingly higher frequen-cies. That is, a stable symmetric equilibrium existed below

    the first anti-symmetric resonance, with a non-symmetric

    equilibrium appearing correspondingly above this resonance.

    III. LUMPED CIRCUIT DISCHARGE MODEL

    A. Circuit model description

    The simulations indicate that r ¼ Ri � ðv01=v11ÞR� 6:3 cm is a transition point that divides the discharge into

    FIG. 2. Fluid results versus r at f¼ 60 MHz for (a) n at the midplane (dotted) and along the bottom (solid) and top (dashed) sheath edges, (b) Te along the bot-tom (solid) and top (dashed) sheath edges, (c) the rf sheath voltage amplitudes Vsh, and (d) the time-averaged sheath widths s at the bottom (solid) and top(dashed) electrodes f¼ 60 MHz case.

    093517-4 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • two distinct regions. The inner region (0 < r Ri, subscript i)has cross-sectional area Ai ¼ pR2i ¼ 0:0124 m

    2 and average

    electron density ni ¼ 3:74� 1015 m�3, and the edge region(Ri < r < R, subscript e) has cross-sectional area Ae ¼ pðR2�R2i Þ ¼ 0:019 m

    2 and average electron density ne¼ 2.27� 1015 m�3. The average electron temperature within bothregions is fairly uniform at Te¼ 4.08 V.

    Figure 6 shows a nonlinear lumped circuit model of the

    discharge with values based on the fluid simulation results.

    The rf current Irf is the sum of the currents going into theinner (i) and edge (e) regions, and the rf voltage Vrf is thepotential difference between the top and bottom electrodes.

    The horizontal branch BD, located at the midplane

    z¼ 1.5 cm of the discharge, divides it into top (t) and bottom

    FIG. 3. Fluid results versus r for the f¼ 80 MHz case for the same diagnostics as in Fig. 2.

    FIG. 4. Fluid results showing contour

    plots at (a) 60 MHz and (b) 80 MHz of

    Iðr; zÞ ¼ 2prH/ which gives the totalcurrent flowing normal to the cross-

    sectional area enclosed by a loop of

    radius r.

    093517-5 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • (b) regions. On this radial branch, the radial current ir¼ it – ibflows from the inner to the edge region of the discharge, giv-

    ing rise to the inductance Lr and resistance Rr of the bulkplasma due to the radial fields. The vertical branches AB,

    BC, AD, and DC show the axial currents and circuit ele-

    ments of the top inner (ti), bottom inner (bi), top edge (te),and bottom edge (be) regions, respectively. For each axialbranch, the C’s are the nonlinear sheath capacitances due to

    the axial fields, which depend nonlinearly on the rf current

    amplitude flowing through them. The L’s and R’s are theinductances and resistances of the bulk plasma due to the

    axial fields.

    The lumped circuit model is in a Wheatstone Bridge33

    configuration, but with some nonlinear elements. The bridge

    circuit is balanced when the potentials at points B and D are

    equal (VB¼VD) so that the radial current ir � it � ib ¼ 0.

    FIG. 5. Fluid results versus r at (a) 60 MHz, (b) 65 MHz, (c) 80 MHz, (d) 85 MHz, (e) 90 MHz, and (f) 95 MHz for Ezs (solid) and Eza (dashed) at four differentphases / ¼ xt ¼ p=4, p=2; 3p=4, and p of an rf half-cycle.

    093517-6 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • Let Zxy be the impedance of the corresponding circuit branchxy, where x¼ t, b indicates top or bottom, while y ¼ i; e indi-cates inner or edge. Then, from the voltage divider rule, the

    circuit is balanced when the ratio of the impedances of the

    top and bottom branches of the inner region is equal to that

    of the edge region

    Zti=Zbi ¼ Zte=Zbe: (11)

    For a discharge in a symmetric steady state, (11) is automati-

    cally satisfied with Zti=Zbi ¼ Zte=Zbe ¼ 1, so that ir andhence Pr/Pe are zero. In a non-symmetric steady state, theabove balance condition is not satisfied, so that the radial

    current ir and hence Pr/Pe are non-zero.We calculate the circuit elements of each axial branch

    by modeling its sheath and bulk plasma regions as uniform

    dielectric slabs with the same cross-sectional area but differ-

    ing thicknesses and dielectric constants. The details of the

    calculation are given in Appendix A, and here we just pre-

    sent the results. The sheath capacitances and bulk inductan-

    ces are

    Cxy ¼�0Aysxy

    (12)

    and

    Ly ¼d

    x2py�0Ay; (13)

    respectively, where x¼ t, b indicates top or bottom, y¼ i, eindicates inner or edge, d � D=2 is the half-width of theplasma bulk, and xpy ¼ ðnye2=ð�0mÞÞ1=2 is the electronplasma frequency in the inner (y¼ i) or outer (y¼ e) region.The corresponding impedances are ZCxy ¼ �j=ðxCxyÞ andZLy ¼ jxLy, respectively. The resistance of each bulk plasmaslab is

    Ry ¼ �TzLy; (14)

    where �Tz is the axial effective collision frequency1 that

    takes into account the electron heating in both the bulk

    plasma and the sheath due to the axial fields.

    We calculate the circuit elements in the radial branch by

    modeling the edge region of the plasma as a dielectric

    between two concentric cylindrical layers with inner radius

    rin ¼ Ri ¼ ðv01=v11ÞR and outer radius rout¼R. Again, thedetails of the calculation are given in Appendix A and here

    we just present the results. The inductance of the cylindrical

    plasma region is

    Lr ¼ln ðv11=v01Þ4p�0dx2pe

    : (15)

    The corresponding impedance is ZLr ¼ jxLr. The resistanceof the cylindrical plasma region is

    Rr ¼ �TrLr; (16)

    where �Tr is the radial effective collision frequency thattakes into account the electron heating in both the edge and

    inner regions of the bulk plasma due to the radial fields.

    Figure 7 shows the fluid simulation results versus f that areused in the circuit model for �Tz (circles), �Tr (triangles), and �m(squares), as well as their linear interpolations. Figure 8 shows

    the fluid simulations results versus applied frequency f for themagnitude of the impedances of the circuit elements in (a) the

    axial branches and (b) the radial branch. No results are shown

    for the unstable cases with f¼ 70 and 75 MHz. From Fig. 8, wesee that in each branch the resistances are much smaller than the

    reactances. Thus, we can neglect the resistances when using

    Kirchhoff’s Laws to solve for the currents and voltages of the

    circuit shown in Fig. 6. In this case, in the sinusoidal steady

    state, applying Kirchhoff’s Voltage Law (KVL) to the top and

    bottom loops of the circuit yields

    0 ¼ j xLi �1

    xCti

    � �it þ jxLrðit � ibÞ

    � j xLe �1

    xCte

    � �ðIrf � itÞ; (17)

    FIG. 6. Wheatstone bridge circuit model of a high frequency, symmetrically

    driven capacitive discharge.

    FIG. 7. Fluid results versus f for �Tz (circles), �Tr (triangles), and �m(squares), as well as their linear interpolations.

    093517-7 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • 0 ¼ j xLi �1

    xCbi

    � �ib � jxLrðit � ibÞ

    � j xLe �1

    xCbe

    � �ðIrf � itÞ: (18)

    For the circuit model, iti ¼ it; ibi ¼ ib; ite ¼ Irf � it, andibe ¼ Irf � ib. Applying the constraint that Pe ¼ Pe0¼ 5 W,as in the fluid simulations, we obtain

    Pe0 ¼1

    2Riði2t þ i2bÞ þ

    1

    2ReððIrf � itÞ2 þ ðIrf � ibÞ2Þ

    þ 12

    Rrðit � ibÞ2: (19)

    Thus, we have three nonlinear algebraic equations (17)–(19)

    to solve for the three unknowns it, ib, and Irf. Since the fluidsimulations assume a Child Law sheath, the sheath widths

    sxy of the sheath capacitances Cxy in the KVL equations arenon-linear functions of the currents as will be shown below

    in Appendix A.

    B. Circuit model comparisons to simulations

    We use the Matlab rootfinding program fsolve toobtain the equilibrium circuit solutions. This program

    requires the choice of some nearby initial conditions (“a

    good guess”) in order to converge to an equilibrium solution,

    if it exists. Three types of initial conditions are used: (1)

    Symmetric initial conditions (s-ic) using the fluid simulation

    results for the 60 MHz case with Irf ¼ 3:19 A. it ¼ ib ¼ 1:37A. (2) Non-symmetric initial conditions (ns-ic) using the

    fluid simulation results for 80 MHz with Irf¼ 2.36 A,it¼ 1.35 A, and ib¼ 0.576 A. (3) Antisymmetric initial con-ditions (as-ic) with Irf¼ 0 A, it¼�ib so that ir ¼ it � ib ¼ 2itand from (19) it ¼ ðPe0=ðRi þ Re þ 2RrÞÞ1=2. Figure 9 showsthe fluid data (circles) and the circuit model results with s-ic

    (solid line), ns-ic (dashed line), and as-ic (dotted line) for (a)

    Irf, (b) Vrf, (c) ir¼ it – ib, (d) Pr/Pe, and the sheath capacitan-ces (e) Cti and (f) Cte; in (e) and (f), the fluid data (triangles)and the circuit model results for ns-ic (dotted-dashed line)

    are also shown for Cbi and Cbe. The pure anti-symmetricequilibrium solution (star) for the circuit is also shown for

    each diagnostic. (Note that there is no stable anti-symmetric

    equilibrium in the fluid simulations.) The solutions on the s-

    ic (solid) line are all symmetric with it¼ ib, while the solu-tions on the ns-ic (dashed) line are non-symmetric from

    f� 66–92 MHz with the degree of non-symmetry decreasingwith increasing f. The fluid data for f¼ 55–65 MHz showgood agreement with the symmetric circuit solutions, while

    those for f¼ 80–90 MHz show good agreement with the non-symmetric circuit solutions. The ns-ic line merges with the

    all symmetric solutions line for f> 92 MHz, while the fluidsimulations still show a slightly non-symmetric steady-state

    at f¼ 95 MHz. The solutions on the as-ic (dotted) line arenon-symmetric and show the approach towards the pure anti-

    symmetric (star) solution. Note that these solutions were

    found to be unstable in the fluid simulations. We also note

    that in the frequency interval f� 66–92 MHz for which thenon-symmetric equilibria exist, the fluid simulations could

    not reach a symmetric equilibrium. The stability of the sym-

    metric equilibria will be discussed further in Sec. III C.

    The frequency fLC at which the inductance Lr of theradial branch of the circuit is in resonance with the effective

    total capacitance of the axial branches of the circuit gives a

    more accurate measure of the anti-symmetric resonance fre-

    quency fa than Eq. (4). The effective capacitance C0xy of each

    axial branch can be found from

    Zxy ¼ j xLy �1

    xCxy

    � �¼ 1

    jxC0xy; (20)

    and the total capacitance of the axial branches is given by

    Ctot ¼C0tiC

    0te

    C0ti þ C0teþ C

    0biC0be

    C0bi þ C0be: (21)

    Then, the anti-symmetric resonance frequency derived from

    the inductances and capacitances of the circuit is

    fLC ¼1

    2pffiffiffiffiffiffiffiffiffiffiffiffiLrCtotp : (22)

    Figure 10 shows fLC from the fluid data (circles) and from thecircuit model results with s-ic (solid line), ns-ic (dashed line),

    and as-ic (dotted line). The pure anti-symmetric circuit solu-

    tion (star) is also shown as well as the line f¼ fLC (dotted-dashed line). As before, the fluid data for f¼ 55–65 MHzshow good agreement with the symmetric solutions line, while

    FIG. 8. Fluid results versus f for the magnitude of the impedances of the cir-cuit elements in (a) the axial branches and the (b) radial branch.

    093517-8 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • that for f¼ 80–90 MHz show good agreement with the non-symmetric solutions line. The pure anti-symmetric circuit

    solution (star) at f� 59.5 MHz lies on the line f¼ fLC. We alsonote that the s-ic line (solid) intersects the f¼ fLC line (dotted-dashed) between f¼ 70 and 75 MHz. This may explain whythe fluid simulation, which also starts from symmetric initial

    conditions, is unstable between 70 and 75 MHz.

    C. Symmetric equilibrium stability

    The analysis of the stability of the discharge equilibria is

    complicated due to the essential role of the small resistive

    impedances and the time variation of the rf period-averaged

    charge hqi within the sheath. The stability analysis is alsoconfounded at the higher frequencies by the increasing influ-

    ence of the second anti-symmetric resonance mode.

    In a purely linear passive circuit, there can be no insta-

    bility, so the instability must be induced by the nonlinear

    dependence of the sheath capacitances on the charge.

    However, the timescale s for the sheath to charge and dis-charge is s � s=uB � 1 ls, with uB ¼ ðeTe=MÞ1=2 being theBohm (ion loss) speed; this implies that s is much greaterthan the rf period. This ordering suggests that the dynamics

    of the rf period-averaged charge hqi plays an essential role in

    FIG. 9. Fluid data (circles) and the circuit model results with s-ic (solid line), ns-ic (dashed line), and as-ic (dotted line) for (a) Irf, (b) Vrf, (c) ir¼ it – ib, (d)Pr/Pe, and the sheath capacitances (e) Cti and (f) Cte; in (e) and (f), the fluid data (triangles) and circuit model results with ns-ic (dotted-dashed line) are alsoshown for Cbi and Cbe.

    093517-9 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • the stability analysis. In Appendix B, we introduce a simple

    “relaxation” form for this dynamics to examine the linear

    stability of the symmetric equilibrium. In addition, the rela-

    tively small reactive impedances of the axial plasma induc-

    tances (13) shown in Fig. 6 are neglected compared to the

    reactances of the corresponding nonlinear sheath capacitan-

    ces. We obtain a cubic equation for the frequency p, withtwo high frequency and one low frequency roots. The high

    frequency roots are always stable. The real (solid) and imagi-

    nary (dashed) parts of the normalized low frequency root

    p/x are plotted in Fig. 11. The symmetric equilibrium is unsta-ble for Re(p/x) > 0. In agreement with the fluid simulations,the symmetric mode is stable for f< 67 MHz and loses stabilityover the frequency range f¼ 67 to 91 MHz which correspondsalmost exactly to that in which the non-symmetric equilibria

    exists, as seen in Fig. 9. As shown in Appendix B, the instabil-

    ity is due to a combination of nonlinear and resistive effects. In

    this model, the symmetric equilibrium is restabilized above

    about 92 MHz. This is in contrast to the fluid simulations,

    which show a weakly unstable discharge above 95 MHz. One

    reason is that the second anti-symmetric resonance, which is

    neglected in the model, becomes significant in this higher

    range of frequencies, and as with the first resonance, disrupts

    the stability of the symmetric mode.

    IV. CONCLUSIONS

    Two radially propagating surface wave modes, symmetric

    and anti-symmetric, can exist in capacitively coupled plasma

    (CCP) discharges. In the former, the upper and lower axial

    sheath electric fields are aligned, while in the latter, they are

    opposed. At high frequencies, the radial wavelengths of these

    modes can be of the order of the plasma radius, leading to spa-

    tial resonances and standing wave effects. For a symmetric

    (equal electrode areas) CCP driven symmetrically, we expected

    to observe only the symmetric mode. However, when the drive

    frequency f is above or near an anti-symmetric spatial reso-nance, we find that both modes can exist in combination and

    lead to unexpected non-symmetric equilibria. We use a fast 2D

    axisymmetric fluid-analytical code to examine a symmetric

    CCP operated in the frequency range of 55–100 MHz at low

    pressure (7.5 mTorr) and low density (�3� 1015 m�3). Thefrequency range encompassed the first anti-symmetric spatial

    resonance fa but was far below the first symmetric spatial reso-nance fs. At lower f, significantly below fa, we found that thesymmetric CCP is in a stable symmetric equilibrium, as

    expected. Typical results at 60 MHz are given in Fig. 2. At

    higher f, near or above fa, a non-symmetric equilibrium appearswhich can be stable or unstable. An example of a stable non-

    symmetric equilibrium is given in Fig. 3 for f¼ 80 MHz. Ascan of frequencies between 55 and 100 MHz at 5 MHz inter-

    vals indicated stable symmetric equilibria at 55, 60, and

    65 MHz, followed by an unstable frequency interval, and then

    stable non-symmetric equilibria at 80, 85, 90, and 95 MHz.

    To understand these results, we developed a circuit

    model, shown in Fig. 6, where the nonlinear axial sheath

    capacitances of the inner and edge regions are connected by

    a radial plasma inductance. The circuit is in the form of a

    Wheatstone bridge, which in the symmetric equilibrium has

    equal currents flowing in the top and bottom axial arms and

    zero current flowing in the radial arm. We calculated the cir-

    cuit elements in Appendix A. Figure 8, showing the magni-

    tudes of the circuit element impedances, indicated that the

    resistances were small compared to the reactances and could

    be neglected in the Kirchhoff’s Voltage Law equations (17)

    and (18). We found good agreement between the fluid simu-

    lation and the circuit model for both equilibria. The model

    indicated that proximity to the anti-symmetric spatial reso-

    nance allows self-exciting of the anti-symmetric mode even

    in a symmetric system. This allows a combination of modes

    to exist, as observed by the non-symmetric equilibria.

    We performed a linear analysis to understand the desta-

    bilization of the symmetric equilibria. In a purely linear cir-

    cuit, there would be no instability, so the instability is

    induced by the nonlinear dependence of the sheath widths on

    the discharge currents and voltages. Both the fluid and circuit

    models assume the non-linear Child Law type sheath model

    derived in Ref. 32, which gives a self-consistent solution for

    a capacitive rf sheath in the low pressure, high frequency

    regime of interest. The analysis, given in Appendix B, is

    complicated due to the essential roles of the sheath capaci-

    tance nonlinearities, the small resistive impedances, and the

    time variations of the rf period-averaged charge hqi within

    FIG. 10. Resonance frequency fLC from the fluid data (circles) and from thecircuit model results with s-ic (solid line), ns-ic (dashed line), and as-ic (dot-

    ted line). The pure anti-symmetric circuit solution (star) is also shown as

    well as the line f¼ fLC (dotted-dashed line).

    FIG. 11. Real (solid) and imaginary parts of p=x versus driving frequencyfor the symmetric equilibrium; the equilibrium is unstable for Re(p/x) > 0.

    093517-10 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • the sheath. We found that the symmetric mode is stable for

    frequencies below the anti-symmetric mode resonance and

    loses stability over the frequency range corresponding to the

    existence of the non-symmetric equilibria as seen in Fig. 9. In

    contrast to the fluid simulations, which show a weakly unsta-

    ble discharge for f > 95 MHz, the circuit model shows thatthe symmetric equilibrium is restabilized for f > 92 MHz.The second anti-symmetric resonance becomes significant at

    these higher frequencies and as with the first resonance may

    disrupt the stability of the symmetric mode. There may also

    be significant plasma density and temperature dynamics,

    neglected in the instability model. Future work could study

    the scaling of the symmetric and non-symmetric equilibria as

    the plasma density is increased. The effect of pressure on the

    equilibrium and stability can also be investigated. Both the

    fluid and circuit models assume azimuthal symmetry and

    neglect the angular or “theta” dependence of the discharge.

    We could extend the present study to 3D in order to examine

    if non-axisymmetric modes can also be excited near their spa-

    tial resonances.

    ACKNOWLEDGMENTS

    This work was partially supported by the Department of

    Energy Office of Fusion Energy Science Contract No. DE-

    SC0001939.

    APPENDIX A: CALCULATION OF CIRCUIT ELEMENTS

    We calculate the circuit elements for the nonlinear cir-

    cuit model of Fig. 6. The elements for each axial branch are

    found by modeling its sheath and bulk plasma regions as uni-

    form dielectric slabs with the same cross-sectional area but

    differing thicknesses and dielectric constants. Then, the

    sheath capacitances are

    Cxy ¼�0Aysxy

    ; (A1)

    where the subscript x¼ t, b indicates top or bottom and thesubscript y¼ i, e indicates inner or edge. The capacitance ofeach bulk plasma slab is given by

    Cy ¼jp�0Ay

    d� �

    x2pyx2

    �0Ayd

    � �; (A2)

    where d � D=2 is the half-width of the plasma bulk and

    xpy ¼nye

    2

    �0m

    � �1=2(A3)

    is the electron plasma frequency in the inner (y¼ i) or edge(y¼ e) region. We note that Cy < 0 so that the correspondingimpedance of this bulk plasma slab is given by

    ZLy ¼�j

    xCy¼ þjx d

    x2py�0Ay

    !: (A4)

    Thus, each bulk plasma slab acts as an inductor with

    inductance

    Ly ¼d

    x2py�0Ay: (A5)

    The resistance of each bulk plasma slab is given by

    Ry �d

    rzyAy¼ �TzLy; (A6)

    where rzy ¼ �0x2py=�Tz is the axial dc plasma conductivityand �Tz ¼ �m þ �sh is the axial effective collision frequency1that takes into account the electron heating in both the bulk

    plasma and the sheath due to the axial fields. Let Psh be theelectron power due to the axial fields in the sheath, and let

    Pbz be the electron power due to the axial fields in the bulk.Then, �Tz ¼ �m þ �sh ¼ ð1þ Psh=PbzÞ�m.

    We calculate the circuit elements in the radial branch by

    modeling the edge region of the plasma as a dielectric

    between two concentric cylindrical layers with inner radius

    rin ¼ Ri ¼ ðv01=v11ÞR and outer radius rout¼R. In this case,the capacitance and impedance of this cylindrical plasma

    region are given by

    Cr ¼2pjp�0ð2dÞln ðrout=rinÞ

    � �x2pex2

    4p�0dln ðv11=v01Þ

    (A7)

    and

    ZLr ¼�j

    xCr¼ þjx ln ðv11=v01Þ

    4p�0dx2pe

    !; (A8)

    respectively. Thus, the cylindrical plasma region acts as an

    inductor with inductance

    Lr ¼ln ðv11=v01Þ4p�0dx2pe

    : (A9)

    The resistance of the cylindrical plasma region is given by

    Rr ¼ln ðrout=rinÞ2pð2dÞrr

    ¼ �TrLr; (A10)

    where rr ¼ �0x2p=�Tr is the radial dc plasma conductivityand �Tr is the radial effective collision frequency that takesinto account the electron heating in both the edge and inner

    regions of the bulk plasma due to the radial fields. Let Pbribe the electron power due to the radial fields in the

    inner region, and let Pbre be the electron power due to theradial fields in the edge region. Then, �Tr ¼ �m þ �i¼ ð1þ Pbri=PbreÞ�m.

    As in the fluid simulations, the sheath widths are given

    by sxy ¼ s0y þ s1xy, where s0y ¼ 2:61kDy is the minimumsheath width and s1xy depends nonlinearly on the rf period-averaged charge in the sheath through the Child Law. Here,

    kDy ¼ ð�0Te=ðehlnyÞÞ1=2 is the Debye length at the sheathedge with hl being the axial edge-to-center density ratio. Weuse hl¼ 0.84 in the model since hl � 0.84 in the fluid resultsover the entire simulated frequency range of f¼ 55 to100 MHz. Then, the sheath capacitances are

    Cxy ¼�0Aysxy¼ �0Ay

    s0y þ s1xy: (A11)

    093517-11 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • For the Child Law capacitance

    CCL;xy ¼�0Ays1xy

    ; (A12)

    the Child Law sheath width is [Ref. 1, Eq. (11.2.15)]

    s1xy ¼ �0KCLV3=4xy

    n1=2y

    : (A13)

    Here, Vxy is the rf voltage amplitude across the sheath, and

    KCL ¼1

    1:23

    0:62

    �0ehluB

    � �1=22e

    M

    � �1=4; (A14)

    with uB ¼ ðeTe=MÞ1=2 being the Bohm speed, and M beingthe ion mass. Introducing the rf period-averaged sheath

    charge hqxyi ¼ CCL;xyVxy and substituting (A13) into (A12),we obtain the charge in terms of the voltage as

    hqxyi ¼Ay

    KCLn1=2y V

    1=4xy : (A15)

    Using this, we can express the Child Law sheath capacitance

    (A12) in terms of hqxyi. Adding the series vacuum capaci-tance, we obtain the total inverse capacitance Dxy of eachsheath

    Dxy ¼1

    Cxy¼ K

    4CLhqxyi

    3

    n2yA4y

    þ s0y�0Ay

    : (A16)

    In the sinusoidal steady state, hqxyi ¼ ixy=x, with ixy beingthe sheath current amplitude.

    APPENDIX B: SYMMETRIC EQUILIBRIUM STABILITYCALCULATION

    The discharge equilibrium and stability in the model are

    determined by applying the time-varying Kirchhoff’s

    Voltage Law around the top and bottom loops in Fig. 6.

    Neglecting the small bulk plasma inductances (13), we

    obtain the two loop equations

    Dtiqt þ Ridqtdtþ Lr

    d2

    dt2ðqt � qbÞ þ Rr

    d

    dtðqt � qbÞ

    þ Dteðqt � qrfÞ þ Red

    dtðqt � qrfÞ ¼ 0 (B1)

    and

    Dbiqb þ Ridqbdt� Lr

    d2

    dt2ðqt � qbÞ � Rr

    d

    dtðqt � qbÞ

    þ Dbeðqb � qrfÞ þ Red

    dtðqb � qrfÞ ¼ 0; (B2)

    with dqt=dt ¼ it; dqb=dt ¼ ib, and dqrf=dt ¼ Irf . From (A16),the inverse capacitances in (B1) and (B2) depend nonlinearly

    (cubically) on the rf time-averaged sheath charges as

    Dti¼K4CLhqti

    3

    n2i A4i

    þ s0i�0Ai

    ; Dte¼K4CLhqt�qrfi

    3

    n2eA4e

    þ s0e�0Ae

    ; (B3)

    Dbi¼K4CLhqbi

    3

    n2i A4i

    þ s0i�0Ai

    ; Dbe¼K4CLhqb�qrfi

    3

    n2eA4e

    þ s0e�0Ae

    : (B4)

    To investigate the stability of the symmetric equilibrium

    in the model, we introduce a simple relaxation form for the

    rf period-averaged sheath charge dynamics

    dhqidt¼ q� hqi

    s; (B5)

    with q and hqi being the sheath charge and its rf period-average, and s being the characteristic timescale for thecharge variation. For the symmetric equilibrium, the top and

    bottom equilibrium quantities are identical, e.g., the zero

    order charges are qt0 ¼ qb0 ¼ q0, the zero order inversecapacitances are Dti0 ¼ Dbi0 ¼ Di0, etc. We linearize theloop equations (B1) and (B2), along with (B5), by assuming

    q ¼ Re½ðq0 þ q1eptÞejxt, with q1 � q0, obtaining

    Di0qt1 þD0i0q0hqt1i þ ðpþ jxÞRiqt1 þ ðpþ jxÞ2Lrðqt1 � qb1Þ

    þ ðpþ jxÞRrðqt1 � qb1Þ þDe0ðqt1 � qrf1ÞþD0e0ðqrf0 � q0Þhqt1 � qrf1i þ ðpþ jxÞReðqt1 � qrf1Þ ¼ 0;

    (B6)

    Di0qb1 þ D0i0q0hqb1i þ ðpþ jxÞRiqb1� ðpþ jxÞ2Lrðqt1 � qb1Þ � ðpþ jxÞRrðqt1 � qb1ÞþDe0ðqb1 � qrf1Þ þ D0e0ðqrf0 � q0Þhqb1 � qrf1iþ ðpþ jxÞReðqb1 � qrf1Þ ¼ 0; (B7)

    and

    hq1i ¼q1

    1þ ps ; (B8)

    with the derivative terms in (B6) and (B7) given by

    D0y0 ¼3K4CLq

    20

    n2yA4y

    : (B9)

    We introduce the symmetric and anti-symmetric perturbations

    qs1 ¼1

    2ðqt1 þ qb1Þ; qa1 ¼

    1

    2ðqt1 � qb1Þ; (B10)

    with the inverse relations

    qt1 ¼ qs1 þ qa1; qb1 ¼ qs1 � qa1: (B11)

    Adding and subtracting (B6) and (B7) and using (B8) and

    (B11), we obtain the equations for the symmetric and anti-

    symmetric perturbations

    DT þDNLð1þ psÞ�1þ ðpþ jxÞðRiþReÞh i

    qs1

    � De0þD0e0ðqrf0� q0Þð1þ psÞ�1þ ðpþ jxÞRe

    h iqrf1 ¼ 0;

    (B12)

    DT þDNLð1þ psÞ�1þ ðpþ jxÞRT þ 2ðpþ jxÞ2Lrh i

    qa1 ¼ 0;(B13)

    with a total inverse capacitance DT ¼ Di0 þ De0, a totalresistance RT ¼ Ri þ Re þ 2Rr, and a nonlinear destabiliza-tion term DNL ¼ D0i0q0 þ D0e0ðqrf0 � q0Þ. The symmetric

    093517-12 Kawamura, Lieberman, and Lichtenberg Phys. Plasmas 25, 093517 (2018)

  • perturbation qs1 in (B12) is always stable. Equation (B13)gives the stability condition for the anti-symmetric

    perturbation

    DT þDNL

    1þ psþ ðpþ jxÞRT þ 2ðpþ jxÞ2Lr ¼ 0: (B14)

    We use the sheath response time

    s ¼ �0ðAiDi0 þ AeDe0Þ2uB

    (B15)

    in (B14), given as the quotient of an average sheath width

    and the Bohm (ion loss) speed, with s � 1 ls. The results areinsensitive to the choice of s for xs� 1. Introducing theequilibrium currents in place of the equilibrium charges

    using i0 ¼ q0=jx and irf0 ¼ qrf0=x in (B14), we have from(B3) that

    DT ¼K4CLji0j

    3

    x3n2i A4i

    þ K4CLjirf0 � i0j

    3

    x3n2eA4e

    þ s0i�0Aiþ s0e�0Ae

    : (B16)

    The destabilization term is given similarly as

    DNL ¼3K4CLn2i A

    4i

    ji0j3

    jx3þ 3K

    4CL

    n2eA4e

    jirf0 � i0j3

    jx3: (B17)

    Equation (B14) is a cubic equation in p with two high fre-quency and one low frequency roots. The symmetric equilib-

    rium is unstable for Re(p) > 0. The high frequency roots arealways stable. The real and imaginary parts of the low fre-

    quency root are plotted in Fig. 11 and show instability over

    the frequency range where the non-symmetric equilibrium

    exists as seen in Fig. 9. A good approximation for the low

    frequency root is found from the linear and constant terms of

    the cubic equation

    p � � DT � 2x2Lr � jjDNLj

    ðDT � 2x2Lr þ jxRTÞs; (B18)

    which gives instability for

    xRT jDNLj > ðDT � 2x2LrÞ2: (B19)

    We see that both nonlinearity and resistive effects are

    required to destabilize the symmetric equilibrium.

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