Collisional sheath dynamics in the intermediate rf

frequency regime

N.Xiang, F.L.Waelbroeck

Institute for Fusion Studies, University of TexasAustin,Texas 78712

email: nongx@mail.utexas.edu

April 16, 2004

Abstract

A sheath model is proposed for the case when the rf frequency ω is

comparable to or larger than the ion plasma frequency of the bulk plasma

ωpi and the ion collisionality in the sheath is significant. In this case, the

ion momentum equation can be solved easily. We find that the ion velocity

in the sheath varies with time and the resulting ion energy distribution is

bimodal even though the rf frequency is much larger than the ion plasma

frequency in the sheath. The results of the model are compared with

the numerical solutions of the fluid equations. Both are in very good

agreement.

1

1 Introduction

Low temperature plasmas are widely used in the semi-conductor industry. In

processing plasma, the ion flux, energy distribution and angular distribution on

the wafer surface are crucial to the applications. These ion properties depend

tightly on the sheath dynamics. Although the majority of plasma processing

is operated at low pressure, rf capacitive discharges at intermediate pressure

(10-200 Torr) have been used extensively for high power gas lasers[1, 2]. Re-

cently, plasma discharges at an atmospheric pressure have attracted growing

attention[3]-[5].

Many theoretical as well as experimental contributions have been made to

the understanding of the sheath dynamics[6]-[15]. It has been found that the ion

dynamics in a collisionless sheath is characterized by the ratio of the rf frequency

ω and the ion transit frequency crossing the sheath ωtr. Usually, ωtr and ωpi

are used interchangeably in the literature. Kawamura et al. have shown that

the ion transit frequency is approximately equal to the ion plasma frequency in

the plasma ωpi for a collisionless sheath[16]. Many collisionless sheath models

in the different rf frequency regimes have been developed[6][10]-[12].

These collisionless sheath models however are only applied to very low pres-

sure discharges where the ion mean free path is much larger than the sheath

thickness so that the ion transit frequency across the sheath is much greater than

the ion collision frequency. For example, in argon discharges, the ion mean-free-

path can be estimated by λi ∼ (330P )−1, where P is the pressure in torr and

λi is in cm[7]. The typical sheath thickness sm is smaller than 1 cm. Thus, the

2

collisions in the sheath are only negligible when the discharge pressure is lower

than 3 mtorr. For higher pressure operations, collisions will play an important

role in determining ion dynamics.

The collisional sheath model is far less understood than the collisionless

model. One possible reason for this is that it is difficult to provide a boundary

condition for a collisional sheath model. For a collisionless sheath, the plasma

sheath boundary is usually defined at the place where the ion velocity equals

the Bohm velocity[17]. However, if the collisions are significant in the sheath,

the Bohm criterion no longer holds[17, 18]. There is no conventional way to

define the plasma-sheath interface.

The presence of the ion collisions adds an extra time scale in the ion dynamic

process. In the high frequency regime, Lieberman obtained an analytical solu-

tion for a highly collisional sheath[7] by assuming that the ions respond only to

the time-average rf field. He found that the sheath capacitance depends only on

the sheath thickness and that a sinusoidal sheath current produces a sinusoidal

voltage across the sum of two sheaths in a symmetric rf discharges. Regimes of

lower collisionality were subsequently investigated numerically in Refs.[8, 19]. In

all the above investigations, the discharge parameters were assumed ω À ωpi, γi,

here γi is the ion collision frequency.

In the practical operations, however, the discharge parameters are often

beyond this range. In an argon plasma with plasma density ni = 5 · 1010 cm−3,

for example, the ion plasma frequency fpi ≈7.4 MHz, The typical rf frequency

f ∼13.56 MHz. If the electron temperature Te ∼ 4ev, for the discharge pressure

3

P ≥ 100 mT, the ion mean free path λi ≤ 0.03cm is a few Debye lengths and

much smaller than the sheath thickness. The ion collision frequency can be

estimated by γi ≥ VB/λi ∼ 10MHz and is comparable to the rf frequency and

the ion plasma frequency.

To our knowledge, little research has been done in this parameter range. In

this paper, we will discuss the sheath dynamics when the rf frequency is compa-

rable to the ion plasma frequency and ion collision frequency γi ≥ max(ω, ωpi).

We will show that the Liberman collisional sheath model[7] is only valid for

the case ω À ωpi ≤ γi. If the collision frequency is significant so that γi ≥

max(ω, ωpi), the ion velocity will vary with time even though the ion density is

nearly time-independent.

The paper is organized as follows. In Sec.II, the sheath model is described.

The ion velocity and ion energy distribution at the electrode are calculated for

(i) constant ion collision frequency, and (ii) constant ion mean-free-path. The

analytical solutions are presented in Sec.III. The comparisons of the analytical

results and the numerical solutions of the fluid equations are shown in Sec.IV.

Lastly, the results are summarized in Sec.V.

2 Model Description

The ion dynamics in both the plasma and sheath can be described by the fol-

lowing cold fluid model.

4

∂ni

∂t+

∂

∂x(nivi) = Rine, (1)

∂vi

∂t+ vi

∂vi

∂x= − e

mi

∂φ

∂x− γivi −Rivi, (2)

Here ni and vi are ion density and velocity respectively, Ri is ionization rate, φ

is the electrical potential, mi is the ion mass. γi is collision frequency.

Since the rf bias frequency is usually much smaller than electron plasma fre-

quency, we will use the drift-diffusion approximation to describe the electrons

dynamics [20, 21]. The continuity equation for the electrons thus becomes

∂ne

∂t+

∂

∂x(neµ

∂φ

∂x−D

∂ne

∂x) = Rine. (3)

Where ne is the electron density, and µ and D are electron mobility and diffusion

coefficient respectively. These two coefficients are related by Te = eDµ , where Te

is the electron temperature .

The system of equation is closed with Poisson’s equation,

ε0∂2φ

∂x2= e(ne − ni). (4)

For the convenience of the numerical solution, we nondimensionalize the Eqs.(1)-

(4) by introducing the following dimensionless quantities,

τ = ωt, z = xd , ui = Vi

VB,

Ni = ni

n0, Ne = ne

n0,

Φ = φTe

, E = −∂Φ∂z ,

ε = λD

d , γ = (γi+Ri)dVB

, ν = RidVB

.

τ0 = dVB

, α = ωτ0, β = d2ωD .

5

Here n0 is the maximum plasma density, λD is the Debye length, d is the plasma

length. (λD =√

ε0Te

e2n0), VB is the Bohm velocity(VB =

√Te

mi), Then Eqs.(1) -

(4) become

α∂Ni

∂τ+

∂(Niui)∂z

= νNe, (5)

α∂ui

∂τ+ ui

∂ui

∂z= E − γui, (6)

α∂Ne

∂τ+

α

β

∂

∂z(−NeE − ∂Ne

∂z) = νNe, (7)

ε2∂E

∂z= Ne −Ni. (8)

The ion and electron continuity equations imply that the total current is con-

served.

ε2∂E

∂τ+

Nui

α+

NE + ∂N∂z

β= I0(τ). (9)

Here I0 is the total current. The first term in Eq.(9) represents displacement

current, while the second and the third represent the ion and electron currents

respectively.

3 Analytical solutions

In the sheath, we define the ion transit frequency ωtr = |∂vi

∂x |, the spatial deriva-

tive terms in the ion continuity and momentum equations thus are propor-

tional to ωtr. In the collision-dominated case, ωtr can be estimated by taking

emi

E ∼ γivi, which gives ωtr ∼ ω2pis

γi. Here ωpis is the ion plasma frequency in

the sheath and ωpis =√

ns

n0ωpi. Here n0 and ns is the ion density in the plasma

center and sheath region respectively(see fig.1). Since the ion density drops

rapidly in the sheath, ns ¿ n0 away from the sheath boundary. We consider

6

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ion densityelectron density

n0 at center

ns(sheath)

z

Figure 1: Schematic representation of the ion density in the plasma and sheath

regions.

the case γi ≥ max(ω, ωpi). Hence, ωtr < ωpis ¿ γi, which means the convec-

tion term in the ion momentum equation can be neglected. The ion momentum

equation thus becomes,

α∂ui

∂τ= E − γui, (10)

Since the ionization rate is much smaller than the rf frequency, the ionization

can be neglected in the sheath. If ωpi is comparable to ω, then ωtr ¿ ω. The

leading term in the ion continuity equation gives

∂Ni

∂τ= 0, (11)

Taking the time average of the first order terms in the ion continuity equation,

we obtain

Niui = Ji = const. (12)

7

We thus have ni ≈ ni(x) + O(ω2tr

ω2 ): the ion density is nearly time-independent.

In the sheath, the ratio of the displacement current and the ion current can be

estimated by Jd

Ji= ε0| ∂E

∂t |enivi

∼ n0ns

γωω2

pi

. If γiω ≥ ω2pi, then Jd

JiÀ 1. This means the

displacement current is dominant in the sheath.

We assume that the total current is sinusoidal so that

ε2∂E

∂τ= −J0 sin τ (13)

Here J0 is constant. Integrating the above equation as in Ref.[6], we obtain the

electric field ,

E =

E0(cos τ − cos ϕ), for s(τ) < z,

0, for s(τ) > z.

(14)

Where E0 = J0ε2 and s represents the electron step front. At τ = ϕ, z = s(τ).

We consider the following two cases.

3-A Constant ion collision frequency γi

Since E = − 1ε2

∫ z

sNi(z)dz, the electron sheath front is governed by

Ni(z)dz

dϕ= J0 sin ϕ (15)

Since Ni(z) = Ji/ui = (γJi)/E and the time-average electric field is E =

E0π (sinϕ− ϕ cos ϕ). Thus,

dz

dϕ=

J0E0

πγJi

sin ϕ(sinϕ− ϕ cosϕ) (16)

8

The sheath voltage is then given by ∆Φ = − ∫ s

zEdz′ = − ∫ π

τE dz

dϕdϕ. By using

eq.(16 ), we obtain for 0 < τ < π,

∆Φ = A0[π

3+

34π cos τ − 1

4τ cos τ − 1

4τ cos τ cos (2τ) +

112

τ cos (3τ) +

38

cos τ sin (2τ)− 19

sin (3τ)] (17)

Here A0 = J0E20

πγJi. Expand ∆Φ =

∑∞k=0 Φk cos kτ . We obtain,

Φ0 = (π3 + 32

27π )A0,

Φ1 = 9πA032 ,

Φ2 = 32A075π ,

Φ3 = −πA096 .

The second harmonic is 15.4 percent of the fundamental, and the third harmonic

is 3.3 percent of the fundamental. For a symmetrically driven discharge, the

phase difference between the grounded and powered sheaths is π. Hence, the

drop across the two sheath Φs ≈ 2Φ1 cos τ . The effective capacitance Csh, given

by Csh = I0dΦsdτ

, is the constant.

The ion energy at the cathode is one of the parameters we are interested in.

Inserting Eq.(14) into eq.(12), we have

ui =

ui(τ0)e−γα (τ−τ0) + E0

α [γα cos τ+sin τ

1+( γα )2

−γα cos τ0+sin τ0

1+( γα )2

+

( γα )2e−

γα (τ−τ0) − E0

γ cosϕ(1− e−γα (τ−τ0))] for s(τ) < z,

ui(τ0)e−γα (τ−τ0), for s(τ) > z.

(18)

In the cathode, ϕ = π. We assume at τ = τ0, ui = ui is the time-average

velocity. Note γα = γi

ω , after a few rf periods, the exponential term can be

9

neglected, The ion velocity at the cathode is then given by

ui

ui= 1 +

γi

ω[

γi

ω cos τ + sin τ

1 + (γi

ω )2] (19)

Setting η = γi

ω and sin θ = η√1+η2

, eq.(19) can be rewritten as

ui

ui= 1 +

η√1 + η2

sin (τ + θ) (20)

Where ui = E0γ .

By eq.(20), we see that only when γi

ω → 0, ui ≈ ui. Clearly, Liberman’s

collisional sheath model is valid only in this limit. If the ion collision frequency

is comparable to the rf frequency, the ion velocity varies with time and the

normalized amplitude of the fluctuation of the ion velocity is less than 1 for any

collision frequency.

At high pressure, elastic scattering dominates over charge exchange collisions,

the ion energy distribution(IED) P (ε) is given by

P (ε) =Ji(τ)Ji

| dε

dτ|−1 (21)

Here Ji =∫ 2π

0Ji(τ)dτ and ε = 1

2u2i is the normalized ion energy.

P (ε) =1ui|dui

dτ|−1 (22)

By eq.(19), we obtain

P (ε) = (γ

E0)2

1√(√

εmax −√

ε)(√

ε−√εmin)(23)

The IED is bimodal with its two peaks corresponding to the minimum and

maximum ion energies given by

εmax =12(u0

η)2(1 +

η√1 + η2

)2, (24)

10

εmin =12(u0

η)2(1− η√

1 + η2)2. (25)

Where u0 = E0α . Clearly, εmax and εmin decrease with the collisional frequency.

The width of the IED is

∆ε =2u2

0

η√

1 + η2. (26)

We see that the IED width decreases with η. If we define the time-average ion

energy ε = 12u2

i , then ε = u20

2η22+η2

1+η2 , thus, ∆ε = 4η√

1+η2

2+η2 ε. For η À 1, the width

is about four times of the ion time-average energy.

3-B Constant ion mean free path λi

For constant ion mean free path γ = ui

λ . Here λ = λi

d is the normalized ion

mean free path. The sheath thickness can be estimated from the collisional

form of the Child law[22], sm ∼ λ4/5Ds λ

1/5i ∆Φ3/5( vB

vs)2/5, here ∆Φ is the sheath

voltage, λDs is the Debye length in the sheath, and vs is the ion velocity in the

sheath. Hence, when λi < ∆Φ3/4λDs, collisions are important.

Eq.(12) then becomes

α∂ui

∂τ= E − u2

i

λ. (27)

If αλ À 1, we can obtain ∂ui

∂τ = 0 and u2i

λ = E. This corresponds to Lieberman’s

model[7]. However, if the rf frequency is smaller and the ion collisions are

significant, so that αλ ≤ 1, the ion velocity is time-dependent.

The electric field is still given by Eq.(14). At the cathode, (27) becomes

α∂ui

∂τ= E0(1 + cos τ)− u2

i

λ. (28)

11

We can show that the solutions for Eq.(28) are periodic(see appendix). Al-

though this equation can not be solved analytically, we can still gain some useful

information. The width of the IED can be obtained immediately by eq.(28),

∆ε = λE0 cos τmax (29)

Here τmax is the time when ui reaches its maximum. If ε represents the time-

average ion energy, then ε = λE0/2. Thus

∆ε = 2ε cos τmax (30)

This means the IED width is less than the twice ion average energy.

Eq.(28) is a first order ordinary differential equation and can be readily solved

numerically. We are interested in the case when the ion velocity in the sheath

is greater than the Bohm velocity. If λi ¿ ωpi

ω λD, then α ¿ ui/λ, the collision

term is then dominant. Thus, we have

u2i = λE0(1 + cos τ) (31)

Hence, the minimum ion energy εmin = 0 and maximum energy εmax = λE0.

The IED is

P (ε) =Ji

Ji

1√1− (1− ε

εmax)2

(32)

Two peaks appear at εmin and εmax.

4 Comparison with the numerical results

To evaluate the assumptions used in the analysis of the previous section, we have

solved Eqs.(5)-(8) numerically for a plasma contained between two electrodes

12

Figure 2: Schematic representation of the plasma and rf circuit model.

and separated from each electrode by a sheath. Fig.2 represents schematically

the plasma and rf circuit model we used for the numerical calculations. The

grounded electrode is then placed at z=0 while the biased electrode is at z=1.

A weak constant ionization rate which is much smaller than the ion collision

frequency is assumed to maintain the plasma. The numerical method and the

boundary conditions are the same as used in Ref.[23].

4-A Calculation for a constant collision frequency

We choose ωωpi

= 0.2 and the bias voltage Φ0 = 100. This is beyond the range

of the Liberman collisional sheath model. Usually, the ion plasma frequency

in the sheath is about one order smaller than the ion plasma frequency in the

bulk plasma. Thus it is expected that ω ∼ ωpis. If the ion collision frequency

13

−0.5 0 0.5 1 1.5 2

−8

−6

−4

−2

0

2

4

6

8

(a) ω/ωpi

=0.2,η=2.5

τ/2π

curr

ent x

10−

3

Ji/α

Je/β

Jd

−0.5 0 0.5 1 1.5 2

−8

−6

−4

−2

0

2

4

6

8

curr

ent x

10−

3

(b) η=10

τ/2π

Ji/α

Je/β

Jd

Figure 3: Numerical results for the ion current, electron current and the

displacement current for ωωpi

= 0.2.

is high enough so that γi À ω, then the ion transit frequency ωtr ¿ ω and

Jd

Ji∼ γω

ω2pis

À 1 and our model is valid. The corresponding results are shown in

figs.2-7.

Fig.3 illustrates the numerical results for the ion, electron and displacement

currents for η = 2.5 and 10. Correspondingly, γi

ωpi= 0.5 and 2 respectively.

We see that the ratio of the displacement current and the ion current increases

with the collision frequency. Note that the ion current varies with time. This

is because the ion velocity depends on time as shown by eq.(19). Clearly, the

larger η is, the better the total current can be approximated by the sinusoidal

function. Even though the displacement current is not described well by a

sinusoidal function in the case η = 2.5, the electric field still can be approximated

14

−0.5 0 0.5 1 1.5 2−100

0

100

200

300

400

500

600

700

800

900

1000

(a) ω/ωpi

=0.2,η=2.5

τ/2π

NumericalE

0(1+cosτ)

−0.5 0 0.5 1 1.5 2−100

0

100

200

300

400

500

600

700

800

900

1000(b) η=10

τ/2π

NumericalE

0(1+cosτ)

Figure 4: Numerical and sinusoidal electric fields at the cathode for ωωpi

=

0.2.

as a sinusoidal function. Fig.4 compares the numerical and sinusoidal electric

fields at the cathode. Again, the larger η is, the better agreement we obtain.

The normalized ion density at the cathode is given in fig.5 for η = 2.5,

5 and 10. It can be seen that the larger η is, the smaller the amplitude of

the fluctuation of the ion density is. This is because the ion transit frequency

decreases as the collision frequency increases. Since the ion density ni ≈ ni(x)+

O(ω2tr

ω2 ), the ion density becomes less dependent of time with a higher collision

frequency. Note that even with a significant time-variation of the ion density,

the ion velocity at the cathode obtained by eq.(19) is still in good agreement

with the numerical result. Fig.6 shows the comparisons betweens the numerical

and analytical ion velocities at the cathode. The agreements are excellent for

15

−0.5 0 0.5 1 1.5 20

0.005

0.01

0.015

0.02

0.025

0.03

Normalized ni(ω/ω

pi=0.2)

τ/2π

η=2.5η=5η=10

Figure 5: Normalized ion density at the cathode for ωωpi

= 0.2.

0 1 20

0.5

1

1.5

2

2.5

τ/2π

u i/u0

(a): ω/ωpi

=0.2,η=2.5

NumericalAnalytical

0 1 20

0.5

1

1.5

2

2.5

τ/2π

(b) : η=5

NumericalAnalytical

0 1 20

0.5

1

1.5

2

2.5

τ/2π

(c) : η=10

NumericalAnalytical

Figure 6: Comparisons of the numerical and analytical ion velocities at the

cathode for ωωpi

= 0.2.

16

0 1 20

0.5

1

1.5

2

2.5

τ/2π

u i/u0

(a): ω/ωpi

=2,η=0.25

NumericalAnalytical

0 1 20

0.5

1

1.5

2

2.5

τ/2π

(b) : η=0.5

NumericalAnalytical

0 1 20

0.5

1

1.5

2

2.5

τ/2π

(c) : η=1.5

NumericalAnalytical

Figure 7: Comparisons of the numerical and analytical ion velocities at the

cathode for ωωpi

= 2.

17

0 0.5 1 1.5 2−200

−150

−100

−50

0

τ/2π

(a) ω/ωpi

=0.2,η=10

NumericalAnalytical

0 0.5 1 1.5 2−200

−150

−100

−50

0

τ/2π

(b) ω/ωpi

=2,η=1.5

NumericalAnalytical

Figure 8: Comparisons of the numerical and analytical sheath voltages for

ωωpi

= 0.2, η = 10 and ωωpi

= 2, η = 1.5.

18

large η. With rf frequency increasing, the ion density becomes less dependent of

time while the ion velocity however varies with time if the ion collision frequency

is high enough.

Fig.7 shows the comparisons of the analytical and numerical ion velocities

at the cathode with ωωpi

=2 for η = 0.25, 0.5 and 1.5. The agreements are very

good for η = 0.5 and 1.5. The discrepancy is obvious for η = 0.25, this is because

the convection term is important in this case. For ωωpi

=2, the rf frequency is

much larger than the ion plasma frequency in the sheath ωpis ( ωωpis

∼15 ). It

is expected that the ion density is nearly time-independent, which is confirmed

by the numerical results. As long as the collision frequency is comparable to or

higher than the rf frequency, however, the ion velocity varies with time instead

of being constant as predicted by the collisional high frequency sheath model[7].

The analytical and numerical sheath voltages are plotted in fig.8 for γi

ωpi=

0.5 and 2. The numerical sheath voltages are obtained by defining the plasma-

sheath interface at (Ni−Ne)/Ni = 0.1. We found that the value of (Ni−Ne)/Ni

used to define the interface does not affect the sheath voltage significantly. The

reasonable agreements between the analytical and numerical results verify the

sheath model.

Lastly, the ion energy distribution is plotted in fig.9 for different ion colli-

sion frequencies(eq.(23)). Note the energy width decreases rapidly with the ion

collision frequency.

19

0 2 4 6 8 10 120

5

10

15

20

25

30

Ion energy

Ion energy distribution for u0 = 5

η=2η=4η=6η=8

Figure 9: The ion energy distribution ( taking u0 = 5 ) for different η.

4-B Calculation for a constant ion mean free path

We assume the constant ion mean free path is related to the gas pressure by

λi ∼ (330P )−1, Fig.10-13 illustrate the results for a constant ion mean free

path. The bias voltage Φ0 = 100.

Fig.10 shows the numerical results for the ion, electron and displacement cur-

rents for different pressures (P = 100 mT and 400 mT, correspondingly, λD

λi≈

0.25 and 0.98). As the pressure increases, the ratio of the displacement cur-

rent and the ion current increases. Although the displacement current is poorly

described by a sinusoidal function, the electric field can be approximated by a

sinusoidal function. The ion density at the cathode is plotted in fig.11 for P =

100 mT, 200 mT and 400 mT. Clearly, the ion density varies with time. The

20

−0.5 0 0.5 1 1.5 2−40

−30

−20

−10

0

10

20

30

40

ω/ωpi

=0.2,P=100mT

τ/2π

curr

ent x

10−

3

Ji/α

Je/β

Jd

−0.5 0 0.5 1 1.5 2−40

−30

−20

−10

0

10

20

30

40

curr

ent x

10−

3

(b) P=400mT

τ/2π

Ji/α

Je/β

Jd

Figure 10: Numerical results for the ion current, electron current and the

displacement current for ωωpi

= 0.2.

amplitude of the fluctuation decreases with the pressure. Note that compared

to the constant frequency case, the ion density at the cathode is higher. For

P = 400 mT, the collision frequency in the sheath is much larger than the rf

frequency, the collisional term is dominant in the ion momentum equation and

the ion velocity is given by eq.(31).

The ion velocities obtained by eq.(31) are compared with the numerical re-

sults in fig.12 for P = 100 mT, 200 mT and 400 mT. The agreement becomes

better for higher pressure. The discrepancies exist for small ui. This is because

the collision frequency is lowered for small ui and the time-derivative terms be-

comes important in this time interval.

For higher rf frequency, the ion density depends less on time and the displace-

21

−0.5 0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Normalized ni(ω/ω

pi=0.2)

τ/2π

P=100mTP=200mTP=400mT

Figure 11: Normalized ion density at the cathode for ωωpi

= 0.2.

0 1 20

1

2

3

4

5

6

τ/2π

u i/u0

ω/ωpi

=0.2,P=100mT

NumericalAnalytical

0 1 20

1

2

3

4

5

6

τ/2π

(b) : P=200mT

NumericalAnalytical

0 1 20

1

2

3

4

5

6

τ/2π

(c) : P=400mT

NumericalAnalytical

Figure 12: Comparisons of the numerical and analytical ion velocities at the

cathode for ωωpi

= 0.2.

22

−0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

τ/2π

u i

(a) ω/ωpi

=2,P=100mT

Numericalby Eq.(2.4.19)

−0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

τ/2π

(b) P=200mT

Numericalby Eq.(2.4.19)

Figure 13: Comparisons of the ion velocities from numerical simulation and

solving eq.(2.5.19) for ωωpi

= 2.

23

ment becomes dominant. The ion velocity at the cathode is described by eq.(28)

for λi ¿ Sm. Fig.13 shows the ion velocity given by eq.(28) is in good agreement

with the numerical results. Again, the ion velocity varies with time.

5 Conclusions

We have presented an analytical solution for the sheath dynamics in the in-

termediate frequency regime when the rf frequency is comparable to the ion

plasma frequency and the ion collisions are significant so that γi ≥ max(ω, ωpi).

In this case, the ion transit frequency across the sheath is much smaller than the

collision frequency and the convection term in the ion momentum equation can

be neglected. Furthermore, the displacement current is much larger than the

conduction current. Our analytical calculations and numerical simulations show

that a sinusoidally biased voltage produces a sinusoidal sheath current in this

parameter range. Hence, the electric field also has a sinusoidal form. The ion

momentum equation at the cathode becomes a first order ordinary differential

equation which is easily solved.

If the ion collision frequency is assumed to be a constant, the ion velocity

in the cathode can be obtained analytically. We have shown that if γi ≥ ω,

the ion velocity varies with time and the resulting IED is bimodal with a width

up to four times of the time-averaged ion energy. If the ion mean free path

is assumed to be constant, the ion velocity can be obtained numerically by

solving the first-order ordinary differential equation. When the ion mean free

24

path is much shorter than the sheath thickness, the ion velocity at the cathode

depends on time and the resulting IED is bimodal with a width up to twice the

time-averaged ion energy.

We compared the results of the model with the numerical results over a

wide frequency range from ω/ωpi = 0.2 to 2, they are in a good agreement.

Our results show that even though the rf frequency is much larger than the ion

plasma frequency in the sheath, the ion velocity varies with time provided the

ion collision frequency is high enough. This conclusion indicates the assump-

tion that the ions respond only to the time-average field in the high frequency

regime[7, 8, 18], is valid only for the case ω À γi, ωpi.

Appendix:

Making the transformation τ = 2T and ui = αλy′τy , Eq.(28) becomes

y′′T + (a− 2q cos (2T ))y = 0. (A1)

Here a = −4 Eo

α2λ and q = 2 Eo

α2λ .

Eq.(A1) is the Mathieu equation. The general solution is[24]

y(T ) = AeµT ϕ(T ) + Be−µT ϕ(−T ). (A2)

Here A, B and µ are constants, ϕ is a periodic function with period 2π. If µ

is pure imaginary, y(x) oscillates aperiodically. Generally, µ has a real part,

thus, after a sufficient long time, the unstable mode is dominant. Takeing

y(T ) = AeµT ϕ(T ) for T À 1, we find

u(τ) = 2αλd

dTlog y = 2αλ(µ +

dϕ

dT). (A3)

Clearly u(τ) is periodic with period 2π.

25

References

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26

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27

Captions:

Fig.1 Schematic representation of the ion density in the plasma and sheath re-

gions.

Fig.2 Schematic representation of the plasma and rf circuit model.

Fig.3 Numerical results for the ion current, electron current and the displace-

ment current for ωωpi

= 0.2.

Fig.4 Numerical and sinusoidal electric fields at the cathode for ωωpi

= 0.2.

Fig.5 Normalized ion density at the cathode for ωωpi

= 0.2.

Fig.6 Comparisons of the numerical and analytical ion velocities at the cathode

for ωωpi

= 0.2.

Fig.7 Comparisons of the numerical and analytical ion velocities at the cathode

for ωωpi

= 2.

Fig.8 Comparisons of the numerical and analytical sheath voltages for ωωpi

=

0.2, η = 10 and ωωpi

= 2, η = 1.5.

Fig.9 The ion energy distribution ( taking u0 = 5 ) for different η.

Fig.10 Numerical results for the ion current, electron current and the displace-

ment current for ωωpi

= 0.2.

Fig.11 Normalized ion density at the cathode for ωωpi

= 0.2.

Fig.12 Comparisons of the numerical and analytical ion velocities at the cathode

for ωωpi

= 0.2.

Fig.13 Comparisons of the ion velocities from numerical simulation and solving

eq.(28) for ωωpi

= 2.

28