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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
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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)
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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)
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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)
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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)
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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)
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(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)
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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)
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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=Pbre�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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