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An analytical model of computing ion energy distributions for multiplefrequencies capacitive dischargesZhuwen Zhou, Rongfeng Linghu, Mingsen Deng, and Deyong Xiong Citation: J. Appl. Phys. 112, 063306 (2012); doi: 10.1063/1.4754447 View online: http://dx.doi.org/10.1063/1.4754447 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v112/i6 Published by the American Institute of Physics. Related ArticlesInvestigation of magnetic-pole-enhanced inductively coupled nitrogen-argon plasmas J. Appl. Phys. 112, 063305 (2012) Separate control of the ion flux and ion energy in capacitively coupled radio-frequency discharges using voltagewaveform tailoring Appl. Phys. Lett. 101, 124104 (2012) The electrical asymmetry effect in geometrically asymmetric capacitive radio frequency plasmas J. Appl. Phys. 112, 053302 (2012) Transition from interpulse to afterglow plasmas driven by repetitive short-pulse microwaves in a multicuspmagnetic field Phys. Plasmas 19, 080703 (2012) Influences of impedance matching network on pulse-modulated radio frequency atmospheric pressure glowdischarges Phys. Plasmas 19, 083502 (2012) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors
An analytical model of computing ion energy distributions for multiplefrequencies capacitive discharges
Zhuwen Zhou,a) Rongfeng Linghu, Mingsen Deng, and Deyong XiongSchool of Physics and Electronic Science, Guizhou Normal College, Guiyang 550018, China
(Received 13 March 2012; accepted 24 August 2012; published online 26 September 2012)
We have developed an analytical multi-frequency model, which results in accurate multi-peaks of
the ion energy distributions (IEDs) for the given control parameters. The model can analytically
calculate and predict the IEDs multiple peaks and energy width with necessary plasma
parameters. This model can also predict the IEDs from multiple frequencies’ drives at any
voltage, the associated results are compared with particle-in-cell (PIC) simulations. The IED at a
surface is an important parameter for processing in multiple radio frequencies driven capacitive
discharges. The analytical model is developed for the IED in a low pressure discharge based on a
linear transfer function that relates the time-varying sheath voltage to the time-varying ion energy
response at the surface. This model is in good agreement with PIC simulations over a wide range
of multiple frequencies driven capacitive discharge excitations. VC 2012 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.4754447]
I. INTRODUCTION
Capacitive radio frequency discharges are widely used
for thin film deposition processing of silicon and surface
modification in the microelectronics industry. The ion energy
distribution (IED) at the substrate surface is an important pa-
rameter in these processes. Since the IED is essentially deter-
mined by the sheath response, it is necessary to establish the
relationship between the control parameters and the sheath
response. It is valuable to develop an analytical model and
particle in cell (PIC) simulations to better understand the
physics of the discharge and more accurately predict the
IED. Multiple frequency drives are often used to control the
width and the average of the IED. The physics of IEDs in
low pressure single frequency capacitive discharges is fairly
well understood.1–3 In some approaches,4–7 simulations
regarding dual frequency ion energy distribution study
sheath parameters and conclude that it is possible to control
the IED by varying the voltage of the low frequency drive.
In another approach,8 an effective frequency concept has
been introduced for dual frequency capacitive discharges to
determine the width and average of the IED. The effective
frequency takes the place of both frequencies and is depend-
ent on the frequencies and voltages of the dual frequency
drive. The IED can then be determined as for a single fre-
quency discharge. Dual frequency discharges have drawn
much attention due to their independent control of the ion
fluxes and ion bombarding energies at the substrates. Semi-
analytical9,10 models were developed to understand the
mechanics in multi-frequency capacitive discharges to some
extent. For both single- and dual-frequencies, research has
been performed by experiments11 and PIC simulations.5 An-
alytical collisional sheath models with charge exchange col-
lisions were developed.12,13 A global model for capacitive
discharges in the collisionless sheath model was proposed to
understand bimodal for IED of single frequency capacitive
discharge.14 However, understanding the mechanics of mul-
tiple frequencies driven discharges is less than adequate. We
have developed an analytical multi-frequency model, which
results in accurate multi-peaks IEDs for the given control pa-
rameters. Incorporating Benoit-Cattin’s model,2 our model
can analytically predict the IEDs from multi-frequency
drives at any voltage with necessary plasma parameters. PIC
simulations are used to verify this model.
II. THEORY ANALYSIS
Benoit-Cattin and Bernard2 analytically calculated the
IEDs and DEi for single frequency drive, in the high-
frequency regime for a collisionless rf sheath, they assumed
(1) a constant sheath width, (2) an uniform sheath electric
field, (3) a sinusoidal sheath voltage, (4) zero initial ion ve-
locity at the plasma sheath boundary. For the high frequency
regime, the ions take many rf cycles to cross the sheath and
can no longer respond to the instantaneous sheath voltage.
Instead, the ions respond only to an average sheath voltage
and the phase of the cycle in which they enter the sheath
becomes unimportant, resulting in a narrower IED. In this
high-frequency regime, DEi was calculated analytically for a
collisionless sheath by Benoit-Cattin et al. and found to be
directly proportional to ~Vsh; �Vsh, and inverse proportional to
xhf , where assume that ~Vsh is sinusoidal sheath voltage am-
plitude, �Vsh is averaged sheath voltage, xhf ; xion are high-
frequency and ion transit radian frequency. Thus, as xhf
increases, the IED width shrinks and the two peaks of the
IED approach each other until, at some point, they can no
longer be resolved (shown in Fig. 1). Therefore, the expres-
sions1,2 for the ion energy distributions function (IEDF or
f(E)), and DEhf at high-frequency (xhf =xion � 1) are
f ðEÞ ¼ 2nt
xhf DEhf1� 4
ðDEhf Þ2ðE� e �VshÞ2
" #�1=2
; (1)a)[email protected].
0021-8979/2012/112(6)/063306/7/$30.00 VC 2012 American Institute of Physics112, 063306-1
JOURNAL OF APPLIED PHYSICS 112, 063306 (2012)
DEhf ¼2e ~Vsh
xhf �shf
2e �Vsh
M
� �1=2
: (2)
For the low-frequency regime (xlf =xion>1;xlf
=xion�= 1): In fact, as for low-frequency parameters flf ¼2MHz; ~Vrf ¼800V, and flf ¼3MHz; ~Vrf ¼800V;xlf =xion¼sion=slf ¼1:73>1, and xlf =xion¼2:60>1. For low-
frequency xlf =xion>1, but not very low frequency
xlf =xion�= 1, the IED for low frequency drive presents a
broad bimodal distribution since DElf / 1=xlf and
DElf >DEhf . The ions cross the sheath in a small fraction of
an rf cycle and respond to the instantaneous sheath voltage.
Thus, their final energies depend strongly on the phase of the
rf cycle in which they enter the sheath. Assume sheath volt-
age is still a sinusoidal voltage for low-frequency
VslðtÞ¼ �Vslþ ~Vslsinxlt, and assume the sheath field is uni-
form. As a result, the IED is broad and bimodal, and the IED
width DEi approaches the maximum sheath drop. The two
peaks in the distribution correspond to the minimum and
maximum sheath drops (shown in Fig. 2). We can derive the
expressions of f(E) and DElf for low-frequency drive under
(xlf =xion>1), which are analogous to Eqs. (1) and (2), refer
to the Appendix A at the end
f ðEÞ ¼ 2nt
xlf DElf1� 4
ðDElf Þ2ðE� e �VslÞ2
" #�1=2
; (3)
DElf ¼2e ~Vsl
xlf �slf
2e �Vsl
M
� �1=2
: (4)
For the multi-frequency (both high and low frequencies)
regime:
Assume a low frequency signal modulated by high
frequency signal will lead a new modulation signal in sheath
Vs ¼ �Vs þ ~Vsð1þ masinxhtÞsinxlt, where xh � xl;ma �1; ma is a modulation factor. Dual frequency drives at vol-
tages (2 MHz–800 V, 64 MHz–400 V) are shown in Figure 3,
and dual frequency drives at voltages (3 MHz–800 V,
64 MHz–400 V) in Figure 4. At multi-frequency drives
FIG. 1. Bimodal IED high frequency (64 MHz, 400 V).
FIG. 2. Bimodal IED low frequency (2 MHz, 800 V).
FIG. 3. The sheath voltage functions at dual frequency (2 MHz–800 V,
64 MHz–400 V), lower frequency (2 MHz) voltage of the sheath is modu-
lated by higher frequency (64 MHz) voltages.
FIG. 4. The sheath voltage functions at dual frequency (3 MHz–800 V,
64 MHz–400 V), lower frequency (3 MHz) voltage of the sheath is modu-
lated by higher frequency (64 MHz) voltages.
063306-2 Zhou et al. J. Appl. Phys. 112, 063306 (2012)
(x1;x2…xm), the sheath voltage function is Vs ¼ �Vs
þ ~Vs
Pm ð1þ masinxhfmtÞsinxlt, e.g., triple frequency drives
at voltages (2 MHz–800 V, 8 MHz–800 V, 64 MHz–400 V)
are shown in Figure 5. These figures show that low-
frequency voltage of the sheath is modulated by high-
frequency voltages, so the IEDs for the lowest frequency
drive at sheath voltage are, respectively, modulated by for
higher frequencies drives at sheath voltages. From Eq. (4),
we can see, for low frequency regime, the IED width DElf is
the maximum when sheath voltage approaches maximum of
peak value, DElf ¼ DEmax, and the bimodal peaks of the IED
for the widest DEmax from the lowest frequency drive are in
accord with the upper and lower limits of DEmax, many nar-
row peaks are distributed between the two limits. Because
there are many times modulated waveform of high frequency
in a low frequency cycle (e.g., Figs. 3 and 4), result in multi-
ple peaks (narrow bimodal peaks of high frequency) appear-
ing many times between broad low frequency bimodal
peaks. About the derivations for multi-frequency regime,
refer to Appendix B at the end. We obtain the expressions
for the IED, f(E), and the energy width DElf ; DEhfm, for
multi-frequency regime are
f ðEÞ ¼ 2nt
xlf DElf1� 4
ðDElf Þ2ðE� e �VsÞ2
" #�1=2
þX
m
!2nt
xhfmDEhfm1� 4
ðDEhfmÞ2
"
�X
n
ðE� ðe �Vs6nDEhfmÞÞ2#�1=2
; (5)
n¼ 0, 1, 2,…N/2.
DElf ¼2e ~Vs
xlf �slf
2e �Vs
M
� �1=2
; DEhfm ¼2e ~Vs
xhfm�shf
2e �Vs
M
� �1=2
; (6)
where xlf is the lowest radian frequency in multi-frequency,
xhfm is a higher radian frequency (xhfm > xlf ). DElf is the
widest ion energy distribution relevant to the lowest fre-
quency, the width center is at e �Vs with upper limit e �Vs
þDElf =2 and lower limit e �Vs � DElf =2. DEhfm is a narrower
energy width relevant to xhfm which distributes between the
upper and lower limits of energy of the lowest frequency.
Subscript m represents multi-frequency times except lowest
frequency, N represents integer ratio of DElf =DEhfm, nt is the
number of ions entering the sheath per unit time, �s is the av-
erage sheath width, and �Vs is the average sheath potential.
We obtain the mean sheath width �s in terms of the mean
sheath voltage �Vs by using the collisionless Child-Langmuir
law
�s ¼ 2
3
2e
M
� �1=4 e0
�Ji
� �1=2
�V3=4s : (7)
The ion current density in the sheath is given by
�Ji ¼ ensuB � 0:61n0uB; (8)
where n0 is the bulk plasma density, ns is the ion density at
the pre-sheath boundary, and uB ¼ ðkTe=MÞ1=2is the Bohm
velocity. This implies1
�s � 2
3
2e
kTe
� �1=4 e0
0:61n0e
� �1=2
�V3=4s : (9)
As the voltage drop across the bulk plasma is relatively small
at low pressures, we let ~Vrf is the fundamental rf voltage am-
plitude applied to the electrodes. The time-averaged sheath
voltage �Vs, which is also the ion kinetic energy per ion hit-
ting the electrode, ei, is given by3
ei ¼ �Vs ¼ ~Vs=2 � 0:8 ~Vrf =2: (10)
We illustrated the multi-peaks of IEDs for multi-frequency
using the analytical model from Eqs. (5)–(10), for dual fre-
quency excitation with (xlf ¼ 2p� 2 MHz; ~Vrf ¼ 800V;xhf ¼ 2p� 64 MHz; ~Vrf ¼ 400V; �s ¼ 0:0113m) in Fig. 6,
and with (xlf ¼2p�3MHz; ~Vrf ¼800V;xhf ¼2p�64MHz;~Vrf ¼400V;�s¼0:0113m) in Fig. 7, triple frequency excita-
tion with (xlf ¼2p�2MHz; ~Vrf ¼ 800V; xhf 1¼ 2p�8MHz;~Vrf ¼800V; xhf 2¼ 2p�64MHz; ~Vrf ¼400V; �s¼0:0145m)
in Fig. 8. In order to the model is more truthful, parameter
0.6 instead of 0.8 in Eq. (10). We also compared the model
with semi-analytical,9 the IEDs peaks for the model are in
accord with semi-analytical in Figures 6–8.
III. PIC SIMULATIONS
The simulations are performed using a one-dimensional
particle-in-cell code XPDP1.15 Starting from the initial con-
ditions, particle and field values are sequentially advanced in
time. The particle equations of motion are advanced one
time step, using fields interpolated from the discrete grid to
the continuous particle locations. Next, particle boundary
conditions such as absorption are applied. The Monte Carlo
FIG. 5. The sheath voltage functions at triple frequency (2 MHz, 8 MHz–
800 V, 64 MHz–400 V), lower frequency (2 MHz) voltage of the sheath is
modulated by higher frequencies (8 MHz, 64 MHz) voltages.
063306-3 Zhou et al. J. Appl. Phys. 112, 063306 (2012)
collision (MCC) scheme is applied for electron-neutral colli-
sions.16 Ion-neutral and neutral-neutral collisions are not
considered here. Source terms, charge density q, and current
density, J, for the field equations are accumulated from the
continuous particle locations to the discrete mesh locations.
The fields are then advanced one time step, and the time step
loop starts over again. The cell size Dx, and time step Dt,must be determined from the Debye length kD, and the
plasma frequency xp, respectively, for accuracy and stabil-
ity: Dx�kD¼ðe0Te=enÞ1=2and ðDtÞ�1�xp¼ðe2n=e0mÞ1=2
.
Here, e0 is the permittivity of free space, e is the elementary
charge, and m is the mass of the lightest species (electrons).
A violation of these conditions will result in inaccurate and
possibly unstable solutions. After the simulation becomes
steady state, the IED is collected over many cycles of the
lowest frequency and normalized to unity. The PIC/MCC
IED results are then compared with the analytical model. In
the simulation, the discharge gap is L¼0.03m, consistent
with typical processing applications. The gap width is di-
vided into 500 cells, max Dx=kD�0:8, and the simulation
time step is Dt¼7:63�10�12, max Dtxp�0:08, argon gas is
used with p¼30 mTorr, np2c¼1E8, initial density of par-
ticles n0¼1�1015m�3. These parameters are chosen in
order to resolve the Debye length and fulfill simulation sta-
bility criteria. The analytical result is shown as the dashed
line, semi-analytical result as the solid line, and the PIC
result is shown as the dashed-dotted line in Figs. 6–8. As can
be seen, the model predicts the 2MHz/64MHz (in Fig. 6)
and the 3MHz/64MHz (in Fig. 7) dual frequency case rea-
sonably well. We have similarly calculated the IEDs for
other dual and triple frequency 2MHz/8 MHz/64MHz cases
and find good agreement. An example of the result for a tri-
ple frequency case is given in Fig. 8. The number of peaks
and their energies are reasonably predicted.
IV. CONCLUSIONS
We have developed the analytical multi-frequency
model, which results in accurate multi-peaks of the IEDs for
the given control parameters, which can analytically calcu-
late and predict the IEDs multiple peaks and energy width
with necessary plasma parameters. The model can also
predict the IEDs from multiple frequencies drives at any
voltage, the associated results are compared with PIC simu-
lations. The IED at a surface is an important parameter for
processing in multiple radio frequency driven capacitive dis-
charges. The analytical model is developed for the IED in a
low pressure discharge based on a linear transfer function
that relates the time-varying sheath voltage to the time-
varying ion energy response at the surface. This model is in
good agreement with PIC simulations over a wide range of
single, dual, and triple frequency driven capacitive discharge
excitations. We developed the model, which is applied both
low-frequency (xlf =xion > 1) and dual-frequency, even
multi-frequency regime (xlf =xion > 1; xhf =xion � 1), but
Benoit-Cattin’s model is merely adapted single high
frequency under xhf =xion � 1. Both Benoit-Cattin’sFIG. 7. Analytical, semianalytical, and PIC IEDs for dual-frequency
3 MHz–800 V, 64 MHz–400 V.
FIG. 8. Analytical, semianalytical, and PIC IEDs for triple-frequency
2 MHz–800 V, 8 MHz–800 V, 64 MHz–400V.
FIG. 6. Analytical, semianalytical, and PIC IEDs for dual frequency
2 MHz–800 V, 64 MHz–400 V.
063306-4 Zhou et al. J. Appl. Phys. 112, 063306 (2012)
single-frequency model and our multi-frequency model
neglect collisional effects in their sheaths, and our model is
not adapted to very low frequency (xlf =xion � 1), which
will be the subjects of future work.
ACKNOWLEDGMENTS
The authors are gratefully acknowledge Professor J. P.
Verboncoeur’s help. This project was supported by the Sci-
ence Foundations of Guizhou Province Education Bureau in
China Grant No. 2010053 and Guizhou Province Science
and technology Bureau in China Grant No. 20122291.
APPENDIX A: DERIVATION OF IED AND DElf
FOR THE LOW-FREQUENCY REGIME(xlf =xion>1;xlf=xion�= 1)
Assume sheath voltage is a sinusoidal voltage for low-
frequency VslðtÞ ¼ �Vsl þ ~Vslsinxlt, and assume the sheath
field is uniform. We can obtain the equation of ion motion
Mdv
dt¼ e
�slfð �Vsl þ ~VslsinxltÞ: (A1)
Integrate Eq. (A1), t0 represents the time an ion entering the
sheath, t represents the time an ion hitting the target, and
assume vðt0Þ ¼ 0
MvðtÞ ¼ e �Vsl
�slfðt� t0Þ �
e ~Vsl
xl�slfðcosxlt� cosxlt0Þ: (A2)
Then we obtain
E
e �Vsl¼ Mv2
2e �Vsl¼ e �Vsl
2�s2lf x
2l M½xlðt� t0Þ�
~Vsl
�Vslðcosxlt� cosxlt0Þ�2:
(A3)
Let a dimensionless parameter Al ¼ �s2lf x
2l M=e �Vsl, we get
E
e �Vsl¼ 1
Al
�x2
l ðt� t0Þ2
2�
~Vsl
�Vslðcosxlt�cosxlt0Þxlðt� t0Þ
þ~V
2
sl
2 �V2sl
ðcosxlt�cosxlt0Þ2�:
(A4)
Integrate Eq. (A2), we obtain
�slf ¼e �Vsl
2M�slfðt� t0Þ2 þ
e ~Vsl
Mxl�slfðt� t0Þcosxlt0
� e ~Vsl
Mx2l �slfðsinxlt� sinxlt0Þ: (A5)
Substituting Eq. (A5) into Al ¼ �s2lf x
2l M=e �Vsl, we get an
expression of Al
Al ¼x2
l ðt� t0Þ2
2þ
~Vsl
�Vslxlðt� t0Þcosxlt0 � a; (A6)
where a ¼ ~Vsl= �Vslðsinxlt� sinxlt0Þ, Eq. (A6) is a quadratic
equation with independent variable xlðt� t0Þ, then we obtain
xlðt� t0Þ¼�~Vsl
�Vslcosxlt0
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðAlþaÞ
p �1þ
~V2
slcos2xlt0
�V2sl2ðAlþaÞ
�1=2
: (A7)
Using binomial theorem to the second item on the right of
Eq. (A7), we can obtain
�1þ
~V2
slcos2xlt0
2 �V2slðAl þ aÞ
�1=2
¼ 1þ~V
2
slcos2xlt0
2 �V2sl2ðAl þ aÞ
� 1
8
� ~V2
slcos2xlt0
�V2sl2ðAl þ aÞ
�2
þ…: (A8)
In fact, as low-frequency flf ¼ 2MHz; ~Vrf ¼ 800V; ~Vsl ¼0:8 ~Vrf =2 ¼ 320V and flf ¼ 3MHz; ~Vrf ¼ 800V; ~Vsl ¼ 0:8 ~Vrf =2 ¼ 320V; xlf =xion ¼ sion=slf ¼ 1:73 > 1, and xlf =xion
¼ 2:60 > 1; ~Vsl= �Vsl ¼ 2, soffiffiffiffiffiffiffiffiffiffiffiffiAlþap
�ffiffiffiffiffiAl
p¼
ffiffiffi8p
psion=3slf
> ~Vsl= �Vsljcosxlt0j ¼ffiffiffi2pjcosxlt0j; j ~Vsl= �Vslcosxlt0j=
ffiffiffiffiffiffiffiffiffiffiffiffiAlþap
< 1. Substituting Eq. (A8) into Eq. (A7), we obtain
xlðt� t0Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðAl þ aÞ
p�
~Vsl
�Vslcosxlt0: (A9)
Substituting Eqs. (A9) into (A4), we obtain
E
e �Vsl¼ 1þ a
Al�
~Vsl
�Vsl
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
Alþ 2a
A2l
scosxltþ
~V2
sl
Al2 �V2sl
cos2xlt:
(A10)
As fl¼ 2MHz 3MHz; Al¼ 26:25 59:23�jaj; jaj � 2;j ~V2
sl=2 �V2slcos2xltj � 2, so a=Al; ~V
2
sl=Al2 �V2slcos2xlt may be
neglected, we obtain
E
e �Vlf� 1�
~Vsl
�Vlf
ffiffiffiffiffi2
Al
rcosxlt: (A11)
Then the energy width is
DElf ¼ Emax � Emin ¼ 2e ~Vsl
ffiffiffiffiffi2
Al
r¼ 2e ~Vsl
xlf �slf
2e �Vsl
M
� �1=2
:
(A12)
Eq. (A12) is the energy width for low-frequency regime,
which is the same as Eq. (4). We also obtain the IED for
low-frequency regime, which equals Eq. (3)
f ðEÞ ¼ dnl
dE¼ dnl=dt
dE=dt¼ nlt
DElf xlsinxlt=2
¼ 2nlt
xlf DElf½1� 4
ðDElf Þ2ðE� e �VslÞ2��1=2:
(A13)
Where using math formula sinxlt ¼ ð1þ cos2xltÞ1=2, and
assuming a change of the ion density in the sheath at low-
frequency is not fast, dnl=dt ¼ nlt � constant.
063306-5 Zhou et al. J. Appl. Phys. 112, 063306 (2012)
APPENDIX B: DERIVATION OF IED AND DEi FOR THEMULTI-FREQUENCY (BOTH HIGH AND LOWFREQUENCIES) REGIME
First, we discuss a dual-frequency (high-frequency
(xlf =xion � 1), low-frequency (xlf =xion > 1)) regime. Sec-
ond, we derive the IEDs for Multi-frequency regime (both
high and low frequencies).
1. Derivation of IED and DEi for the dual-frequencyregime (high-frequency (xlf =xion � 1), low-frequency(xlf =xion>1))
We know that a low frequency signal modulated by high
frequency signal will lead a new modulation signal in sheath
Vs ¼ �Vs þ ~Vsð1þ masinxhtÞsinxlt, where xh � xl;ma �1; ma is a modulation factor. Then we get an equation of ion
motion in sheath
Mdv
dt¼ e
�s½ �Vs þ ~Vsð1þ masinxhtÞsinxlt�
¼ e
�sð �Vs þ ~VssinxltÞ þ
e
�s~Vsmasinxhtsinxlt:
(B1)
Comparing Eqs. (B1) with (A1), we find the first item on the
right of Eq. (B1) is a force related low-frequency regime ion
motion equation, which still follows the law of the IED of the
low-frequency regime. The second item ~Vsmasinxhtsinxlt is
low-frequency voltage modulated by high-frequency voltage,
of which modulated amplitude is ~Vm ¼ ~Vsmasinxht, the am-
plitude of low frequency voltage is a sinusoidal function fol-
lowing the law of high frequency voltage, the modulated
wave is sinxht, the carrier wave is sinxlt (e.g., shown in Figs.
3 and 4). There are N times modulated waveform in a low fre-
quency cycle, if xh=xl ¼ srfl=srfh ¼ N. We note that
DEi ¼ 2e ~Vs=x�sð2e �Vs=MÞ1=2;DEi / 1=x, or DEi / srf ;DElf
> DEhf ;DElf =DEhf ¼ N. So we find there are N times narrow
bimodal peaks for high frequency distributing between wide bi-
modal peaks for low requency, so we obtain the first modulated
bimodal IED for high-frequency with energy center at e �Vs
f0ðEÞ ¼2nt
xhf DEhf1� 4
ðDEhf Þ2ðE� e �VsÞ2
" #�1=2
: (B2)
The second and third modulated bimodal IED for high-
frequency with energy center at e �Vs6DEhf
f61ðEÞ ¼2nt
xhf DEhf1� 4
ðDEhf Þ2ðE� e �Vs6DEhf Þ2
" #�1=2
:
(B3)
The fourth and fifth modulated bimodal IED for high-
frequency with energy center at e �Vs62DEhf
f62ðEÞ ¼2nt
xhf DEhf1� 4
ðDEhf Þ2ðE� e �Vs62DEhf Þ2
" #�1=2
;
(B4)
and so on. The ðN � 1Þth and Nth modulated bimodal IED
for high-frequency with energy center at e �Vs6nDEhf
f6nðEÞ ¼2nt
xhf DEhf1� 4
ðDEhf Þ2ðE� e �Vs6nDEhf Þ2
" #�1=2
:
(B5)
Then we can deriver the multi-peaks IEDs for high-
frequency with N times modulated bimodal peaks
fhf ðEÞ¼2nt
xhf DEhf1� 4
ðDEhf Þ2X
n
ðE�ðe �Vs6nDEhf ÞÞ2" #�1=2
:
(B6)
So all the IEDs for the dual-frequency regime(high and low
frequencies)
f ðEÞ¼ flf ðEÞþ fhf ðEÞ¼2nlt
xlf DElf1� 4
ðDElf Þ2ðE�e �VslÞ2
" #�1=2
þ 2nt
xhf DEhf1� 4
ðDEhf Þ2X
n
ðE�ðe �Vs6nDEhf ÞÞ2" #�1=2
;
(B7)
where n¼ 0, 1, 2,.. N/2.
DElf ¼2e ~Vsl
xlf �slf
2e �Vsl
M
� �1=2
¼ 2e ~Vs
xlf �slf
2e �Vs
M
� �1=2
;
DEhf ¼2e ~Vsh
xhf �shf
2e �Vsh
M
� �1=2
� 2e ~Vs
xhf �shf
2e �Vs
M
� �1=2
:
(B8)
We assume ~Vsl ¼ ~Vs; ~Vsh � ~Vs in Eq. (B8), because ~Vs
derives from low frequency voltage amplitude in Eq. (B1),
and high frequency voltage amplitude is ma~Vs from Eq.
(B1), jmaj � 1, if good modulation, jmaj � 1.
2. Derivation of IED and DEi for the multi-frequency(both high and low frequencies) regime
Assuming that there are multiple frequencies voltages on
the sheath, but we think there are only the lowest frequency
voltage with the widest energy spread in multiple frequencies,
the others (whether high frequencies or low frequencies) are
all modulated voltages for high frequencies. If all high fre-
quencies modulated signals areP
m~Vssinxhfmt, and the low-
est frequency carrier wave is ~Vssinxlt, the expression of the
sheath voltage for the multi-frequency regime
Vs ¼ �Vs þ ~Vs
Xm
ð1þ masinxhfmtÞsinxlt: (B9)
Then, the equation of ion motion in sheath for the multi-
frequency regime
Mdv
dt¼ e
�sð �Vs þ ~VssinxltÞ þ
Xm
e
�s~Vsmasinxhfmtsinxlt:
(B10)Comparing Eqs. (B10) with (B1), we find there are more
high frequencies modulated signals (m times) in multi-
frequency regime, these high frequencies voltages all
063306-6 Zhou et al. J. Appl. Phys. 112, 063306 (2012)
modulate one the lowest frequency voltage, both high and
low frequencies have a semblable bimodal IED (shown in
Eqs. (1) and (3)), and these bimodal peaks of high frequen-
cies IEDs are all distributed between the upper and lower
limits of the bimodal IED of the lowest frequency. Accord-
ing to the same methods as the derivation of the dual-
frequency (both high and low frequencies) regime in the Sec. I
of Appendix B, using Eqs. (B1), (B7), and (B10), and addingXm
to before the second item in Eq. (B7), we can obtain
the expression of IEDs of multi-frequency regime
f ðEÞ ¼ flf ðEÞ þX
m
fhfmðEÞ
¼ 2nt
xlf DElf½1� 4
ðDElf Þ2ðE� e �VsÞ2��1=2
þ�X
m
� 2nt
xhfmDEhfm
"1� 4
ðDEhfmÞ2
�X
n
ðE� ðe �Vs6nDEhfmÞÞ2#�1=2
; (B11)
n¼ 0, 1, 2,. N/2.
DElf ¼2e ~Vs
xlf �slf
2e �Vs
M
� �1=2
;DEhfm¼2e ~Vs
xhfm�shf
2e �Vs
M
� �1=2
:
(B12)
Equations (B11) and (B12) are all the same as Eqs. (5)
and (6).
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