Modelling

The electron sheath edge motion is a strongly nonlinear function of time and is described by the instantaneous sheath position s(t). Ions reach the sheath edge atx =0

(3.1) with the Bohm's velocity Us

=

~Te /mj and hit the electrodes at x

=

^sm where sm is the maximum sheath width. The bulk plasma is assumed to have a density of

no

and electron temperature of Te ; the electron and ion densities within the sheath are ne and ni. respectively. In this model it is assumed that current source has multiple RF frequencies applied to the electrode. Due to the RF source an oscillating sheath will appear near the electrode. Current density l(t) passing through the sheath is a superposition of different sinusoidal RF frequency components and can be given as,

N

l(t)

=

lij sin((Oijt)

+

I,lk sin((Okt+ek)

k=!

Depending on the number of RF frequencies used in Eq. (3.1) defines the wafeform of sources current, where Ik' (Okand ek are 'current amplitude, source frequency and phase angle of k-th frequency, respectively. The one-dimensional spatiotemporal variation of the ion density, ni (x, f), the ion drift velocity, ui (x,t), the electric potential, V(X,f) and the instantaneous electric field, E(x,t) are described by the ion continuity equation,

and ion momentum balance equation,

(lUi +u. (lUi =~E

(It ^I (lx mi

(3.2)

(3.3)

1

where, mi is the ion mass and e is the charge of electron. For plasma sheath, crucial parameter determining the dynamics of plasma sheath is (0/ (Oi where, (0is the RF frequency and (OJ =2rc /'oi,is the ion oscillation frequency within the sheath and 'Oi is the ion transit time. We can specify two distinct regions in collisionless sheaths [12],

(0

»

(OJ: The ion density is time independent, and ions respond only to the time- average electric field.

(0« (Oi:The ion density is time dependent, and ions respond instantaneously to the electric field.

38

For OJ» OJ;, since ions respond only to the time average electric field, it is assumed in all time independent models that, an;

=

0 in Eq. (3.2) and (3.3) [3]-[6], [43], [45]-[49].

at

According to Jiang et ai. [12] for DF-CCP devices of practical interest where a low- frequency source of I or 2 MHz is used, the inequality of OJ» OJ; ion will not hold, especially, when light gases are use. In this case, the condition, OJ» OJ;, ion is conserved for the high-frequency source, while at the same time we have ^OJ« OJ; ion for a low-frequency source. In order to control the ion energy by a low-frequency source, the low-frequency voltage Vlow is much larger than the high-frequency voltage

Vh;gh. Thus, the sheath properties are dominated by the low-frequency source. The time-

independent models may not be appropriate for a DF-CCP since the ion density and velocity are now time dependent.

Edelberg et ai. [49] treated this problem in a more general way. They nondimensionalized the eq. (3.2) and (3.3) by using the dimensionless parameters,

T =ft

_,

maximum sheath potential and Umnx is the maximum velocity at which the ions enter the

[ J

^{1I4 3/4}

sheath and deis

= '!:~

^{2e V}

m";'/2

is the Child-Langumiar sheath width, where

3 m; J;

J;

=

^enOuB is the ion current density. Using these parameters in Eqs. (3.2) and (3.3) we have,

J

and,

an; on;!i;

⁰

y-+--=

at ax

aU; _ au;

^e-

y-+u.-=-E

at 'ax

^m;

(3.4)

(3.5)

Here, y is the term that defines the importance of time dependent parameters of Eq.

(3.4) and (3.5) and can be express as,

y=[ ^f YV

^max

J

^1/4

3n

f

^p;}. 2

(3.6)

[2 J1I2

where,

f

pi

=

_1_ e no is the ion plasma frequency.

271: eOmi

In collisional case, the typical value used for a low frequency wlf=IO MHz in AI plasma where the plasma density is no

=

2x1010 cm-3 and Vmax is 70 V. Using these values in Eq.

(3.6), the value of

r

is found to be 0.05 that is very less than unity. Thus, we can neglect the time-dependent terms in Eqs. (3.4) and (3.5). However, still the ion density and electric field will be a function of time. Assuming no ionization or recombination within the sheath ion particle conservation law gives [6], [12],

(3.7) According to Boyle et al. [6], in the intermediate-pressure range, when the energy gained between two subsequent ion-neutral collisions is larger than the ion thermal energy -ksTi, the dominant momentum transfer process is due to charge exchange (cx) and gas-kinetic-type collisions (k) with the parent gas, with a cross-section ^CTcx and ^CTk virtually independent of ion velocity. These two cross sections are generally incorporated into a constant single hard-sphere cross section CT = CTcx +^CTk. A constant mean free path model is therefore appropriate.

This model is also known as the variable mobility model. If we define the mobility as, fli (Ui) =Ui / E then the ion mobility beocomes,

(3.8)

which, itself a function of the ion drift velocity, Ui. For argon, the cross section for charge exchange and kinetic repulsion are equal to CTcx

=

5xlO-¹⁵cm² and CTk

=

4.2xIO-¹⁵ cm^2, so that the ion mean free path is

Ai

⁼11300p ,where Ai is in centimetres and p is in Torrs [6]. We note here that the value of

Ai

is always less than the value of sheath width, Sm in a collisional sheath that is considered in this model. However, in a collionless sheath that is considered in Jiang's model [12],

Ai

is always larger or equal to the sheath width. Eq. (3.8) gives, 2eAiE(X,t)

ui(x,t)

=

40

(3.9)

Now from (3.7) and (3.9) we have,

71Jni

2eAiE(x,t) (3.10)

where, UB

=

~Te1m; is the Bohm's velocity. Poisson's equation for instantaneous electric field E(x, t)within the sheath can be given as,

x> s(t) x <s(t)

(3.11)

J

Proper boundary conditions at the plasma sheath interface can be defined consistently following Riemann's work [45], [46]. However, one must solve the above equations by a numerical method. For a full analytical solution, we adopt an inconsistent condition here. Riemann [46] has also shown that the inconsistent boundary condition can give good approximations of this problem; i.e., E(s,t) =0 used by Lieberman [47], [48] for p =JOOJpiIJ/LJ

=

5 and E =Te I eAo used by Godyak [43] for P =J oOJp; lJiOJ

=

0.5.

Here, we only consider the electron-free ion sheath. Franklin [50], [51] has shown that electric field in the electron-free ion sheath is comparable to E

=

TeI ell.o, theoretically and experimentally. This value is modified by Riemann [45], [46]. As discussed above, the ion flow is in a steady state condition; thus Eq. (3.4) turns into a(m;u;212) =Te 121l.o =eE. The electric field at the sheath boundary (x = s) is then given by,

E(x,t) =_e_;T 2eAo

x ~ s(t) (3.12)

-; According to Franklin [50], [51], as the plasma becomes collisional the density at the 'plasma edge' falls, the corresponding Debye length increases, and at some point ions undergo at least one collision in traversing the 'transition region' and they consequently

do not pass through the Bohm speed. Then the structure of 'plasma-transition layer- collisionless sheath' gives way to that of 'collisional plasma-collisional sheath'. To avoid complexity we are considering that the ions enter at the sheath at Bohms velocity.

So we will consider a transition layer i.e. presheath. At the 'patch point' of plasma- sheath the error in calculation is tolerable in some quantities if we assume that,

E(x t)=_eT .

) eAD' x> s(t) (3.13)

Here, the electron debye length, AD = ~eoTe / noe2 . It is notable that for time independent collisional case Boyle et al. [6] assumed that E(O, t) = O. For boundary condition we assumed that,

From current conservation law,

V(O,t)=O (3.14)

(15) where, I^d is the displacement current,

h

and Ie are the ion and electron conduction currents, respectively. As for the electron-free zone, the current is created by the variation of electric field. Thus, the electron and ion conduction currents can be neglected. The total current is dominated by the displacement current that is equal to the total RF source current [12], [48]. Thus,

(3.16)

-I.

For calculating the potential and sheath width introduction of dimensionless parameters will help. We assume that, e;=X/AD' 1]=S(t)/AD' OJ=OJlj' I=Ilj' r=OJljt,

ak

=

OJk/ OJlj' f3k

=

I k / I lj .Also the dimensionless plasma potential, ion velocity and

. d . . . b m( ) eV(x,t) ( ) u;(x,t) n;(x,t)

IOn ensltYlsglven y, ^'¥

e;,r

=---, u

e;,r

=---, n(e;,r)=~--.

Te UB

no

From Eq. (3.16) we can show that,

d1]

= So

[sin(OJt)+ if3k Sin(akt+ek)]

d(r) nee;, r)AD k~l

(3.17) Where,

So = --

I is the effecting sheath oscillation motion amplitude. From Eq. (12)

enoOJ we have,

which implies, for

e;

^<1](r)

Also from Eq. (3.11),

E(e;,r)

= -

1 for 2

E(x, t)

=

^_eT ^E(C;,r)

eIlD

42

e;

<1](r) (3.18)

(3.19)

Again from Eq. (3.18) we have,

(3.20)

1 d<P E(e;,T)=2=- de;

Integrating Eq. (3.21) we find,

For

e; =

^1](T) we have,

Using Eqs. (3.10) and (3.18) we obtain the expression for ion density as,

(3.21)

(3.22)

(3.23)

(3.24)

So inserting Eq. (3.24) in Eq. (3.17) we have,

!!YL =

^So [sin(wt) +

f ^1\

^{sin(akt +Ok)] (3.25)}

d (T) n(e;, T)A.O k=!

Integrating above equation,

=}1](T)

=

^So

[1-

^{COS(T) +}

f

^13k(cosOk - cos(akt +Ok)] (3.27)

nee;,T)A.O k=l ak

Now, the sheath motion can be given as set)

=

^1]

(T)A.o .

From Eq. (3.22), we see that the sheath motion is a function of ion density, ni. In this model, ni is a function of instantaneous electric field given by Eq. (3.10).

To find the value of electron-free ion sheath potential, at any point in the sheath, we will consider,

e;

>^1](T). Then from Eq. (3.13) we have,

d<P

E(e;,r)=I=- de; (3.28)

which implies that,

From Eq. (3.11) we have,

E(x,t)

=

^_eT ^E(r;,r) eAD

(3.30) .

a

²(eV(x,t)/Te)

=;----~=

a(x/ AD)2

Assuming, r;⁼71 ,

a2tP(r;,r)

=; 2 -

-nCr;,

r)

=

H

c

(let)

ar;

(3.31)

(3.32)

i

App ymgI. the boun ary cond d't'1IOns atP(r;,r) - -.05 t0 ---atP(r;,r) __ atP(r;,r) and

ar; ar; ar;

'th f . a atP(r;,r) a h

usmg e trans ormatIOn -

= ----

we ave,

ar; ar; atP

atP(r;,r) a (atP(r;,r))=H

ar; atP ar; c

=;

ra'!>/a,

atP(r;,r) a(atP(r;, r))

= tV)

^{H catP}

-0.5 ar; ar; <Pn

(3.33) Now multiplying Eq. (3.32) and Eq. (3.33) we have,

atP(r;,r) -~2Hc(tP(r;,r)-tP(71,r))+1/4x So [sin(ll)t)+

Ifh

sin(akt+8k)]

ar ^nCr;,

^{r)AD k=!}

(3.34) Integrating Eq,(3.34) we obtain,

(PCp) atP(r;,r)

J(/) -=========== =71(r)

'!>ry ~2Hc(tP(r;,r)-tP(71,r))+1/4

=;~2H ctP(r;,r) +ary - ~2H ctP(71,r) +ary

=

71(r)

44

(3.35)

.•..

.-

where,

an =

114 - 2H c<P(T/,7:). From here, the value of <P((,r) can be determined, and the spatiotemporal variation m sheath potential can be obtained. Using

<P(C;,7:)

=

eV(x,t)/Te we have the time dependent expression of sheath voltage V(x,t).

The sheath potential <P is related to the sheath motion given by Eq. (3.35). Thus, sheath voltage is also a function of instantaneous electric field.