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)zhuwenz@gmail.com.

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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