Investigation of a CF4/Ar RF gas discharge

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Investigation of a CF4/Ar RF gas discharge

Citation for published version (APA):

Vallinga, P. M., Meijer, P. M., Smits, A. J., & Hoog, de, F. J. (1987). Investigation of a CF4/Ar RF gas discharge. In K. Akashi, & A. Kinbara (Eds.), ISPC 8 : International symposium on plasma chemistry, Tokyo, August 31 -September 4, 1987, vol. 1

Document status and date: Published: 01/01/1987

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АВSТRАСГ

INVESТIGATION OF А СF₄/Аг RF GAS DISCНARGE

Р.М. Vallinga, Р.М. Meijeг, A.J. Smits, and F.J. de Hoog, Eindhoven Univeгsity of Technology, Depaгtment of Physics,

Р.О. Вох 513, 5600 МВ Eindhoven, The Netheгlands.

The Boltzmann equation has been solved numeгically in the bulk of а low pressuгe (0.05-1 tогг) СF₄/Аг RF etching plasma Ьу using the multiple time scales foгmalism. Using the гesul ting electгon eneгgy distгibution function fог evaluation of the electгon paгticle and eneгgy Ьalance, the maintenance value of the effective electгic field, the electгon chaгacteгistic eneгgy and

the absoгbed роwег рег electгon have been calculated.

In огdег to obtain а complete description of an RF plasma, а theoгy of the electгic sheaths at the electгodes is included. Consideгing а collisionless sheath and using some appгoximations, we calculated the sheath potentials, electron and ion density pгofiles and the time dependent behaviouг of these pгopeгties.

INTRODUCГION

We consideг а СF4/Аг gas dischaгge between two planaг electгodes and we assume that the fгequency of the electric field, w (w/21r = 13.56 МНz), satisfies W; ~ w

<

We and WTem ~ 1, where w1 and We аге the ion and electгon plasma fгequencies гespectively, and Tem is the time fог electron molecule/atom collisions.

BULK PROPERTIES

In огdег to descгibe the pгopeгties of the bulk гegion of an RF dischaгge, the knowledge of the electгon distгibution function f is indispensaЫe. Hence, we consideг the electгon Boltzmann equation

(1) wheгe V г is the gradient орегаtог in configuгation space and Vv that in velocity space,

Jen

гepгesents the elastic e-n (electгon-neutгal) collisions, while

Ien

descгibes the total inelastic e-n collisions. Неге, we neglected the electгon-electгon and the electгon-ion collision teгms because of the low ionization degгee. In огdег to solve this equation, the small рагаmеtег

~ = (т/М)114

, wheгe m and М denote the electгon and molecule masses

гespectively, is used as an expansion рагаmеtег fог the electгon distгibution function [1], i.e.

(2)

(3)

introduced and the different terms of the Boltzmann equation аге labelled with

Т], Т]2

etc. according to their estimated value. Rearгanging the terms and taking together those of the same magni tude, wi th ТJ serving а bookkeeping role, а set of equations is obtained for

fo , f 1 ,

and

f 2

[1]. Higher order terms need not Ъе considered. From these equations it appears that after many elastic col l isions ( i. е. the fastest time scale т ₀➔ 00 ) the zeroth order

distribution function becomes isotropic in velocity space and f1 , f2 сап Ье

expressed in terms of

fo ,

yielding

(3)

where

0 _f1

₌

Tem(u)Л

fo

.

1 f1=

₁

₊

Tem(u} _jWТem(u)

!!

fo

.

(4а)

1 f2

=

_Зj<,)1 2

[

А· Tem(u} v2

!!+

!!•v2Tem{u)Л ]

fo

1

+

j<,)Tem{u)

.

V {4Ь) and 1 Tem(u}

2

2 f2

=

_12j<,)_{v 2}

[ !!·

₁₊ _j<,)Tem(u)V

!! ]

fo

(4с)

These expгessions, where the differential operators Л,

!!

are defined as

е а е а

А

= -

Е₀(г)-

₈

- vVг • and В

= -

Е₁-

₈

.

- т- - V - т- V (5)

аге obtained if the electric field is assumed to Ъе of the form

Е Е

( ') Е j<,)To

_ : _о

r

+ _1е • (6)

The dc space-charge field

Io(r)

varies with position, but

g

₁ is assumed to Ье constant. The

r

dependency has not been pointed out in Eq.(3). From Eqs. (3) and (4) we сап conclude that

f

2 gives а small time dependent isotгopic

contribution which vaгies wi th the douЫe fгequency 2ы. А dc part of the electric field and/or spatial inhomogenei ties wi 11 also give rise to an isotropic рагt which varies with the ground frequency <,). Finally we make the

separation

fo

=

n(r)h(v)

and take together the diffusion and the space char~e field term Ьу introducing an effective diffusion length Ле f [1], (Л/Ле f) =

Da/De, which is larger than the cavi ty diffusion length А. {The meaning of Da/De will Ье discussed later on.) The dc equivalent of the isotropic part of

the second ordeг equation in terms of the energy variaЬle

u

= mu2/2e yields

(7) f ( ) --

2тrh(u)(2me)з12,

Uv= U + Uv, V Е2 1

Е

2

= {l

~

2 ) , Q _ TemNu,

jf(u)u

112du = 1

(noгmalization

condition),

+<,)Тет em О

(4)

Qi

-

ionization,

Qe-

excitation,

Qa-

attachment. and

Qv-

vibгation, ~

=

рагаmеtег which descгibes the distгibution of eneгgy, afteг

ionization, between the ргimагу and secondaгy electгon, 0.5

S

~

S 1

,

и1= ionization eneгgy,

Ue=

excitation eneгgy,

Uv=

vibгation eneгgy.

Неге, the vibгation teгm has been consideгed as an exci tation pгocess, а

slight adjustment to the expгession used in [1].

If Tem and the cross sections

QJ

are constant in velocity space, analytic

solutions can Ье calculated [1]. Now however, we consider the actual energy

dependent cгoss sections (see ТаЫе 1) and will only present numerical гesults

(fог ~

=

0.5). Integrating Eq.(7) оvег all energ'J space we obtain the electгon paгticle balance equation

(8)

2е ½ t»

N(-) JUf{u)Q1 ,a(u)du .

т о

This equation гelates

E/N

to NЛef, as can Ье seen in fig.l. Тhese аге the

maintenance values of the electгic field inside the plasma, because we have

substituted Dа/Л2

= De/(Aef)

2, wheгe

Da

is the coefficient of ambipolar

diffusion which occurs in the steady state discharge, and

De

is the

coefficient of fгее diffusion. Тhese coefficients are related accoгding to

Da=

Dem<v

e

m>l(M<v

1

m>),

wheгe (v1

m>

is the aveгaged ion-molecule/atom collision

frequency. This appгoximation is only justified for high electгon densities.

i.e. nеЛ2 ~ 1012m-1 [2]. In the actual RF plasma, as used fог plasma etching,

we геасh this limi ting value of nеЛ2 .[З]. The eneгgy Ьalance equation is given Ьу

Е

2

_J;i!a

fX> ,,002 1 fu2_f

(и)

~ N

т

2

)312

=

~ UJJuQJ (u)f(u)du

+

JU Qa(u)f(u)du

+

ЗN2А2 Q ( )du ,

(9)

пе т е J о о е f em u о

~

10-0

-ф , ,

<

, tn 1 ,.. - Аг n ₃ ---CF 4 CJ

-10-9 о 2 4 6 8 10 о 4 8 12 16 20 24 28 EIN (х 10-16Ycm2)

(5)

From this equation we obtain information, as shown in fig.2 and 3 respectively, about the power absoгbed рег electгon, defined Ьу 0 = ½aE2/ene,

and the гelative роwег consumed Ьу the diffeгent pгocesses. (Тhе eneгgy

absorbed Ьу vibгational excitation was less than 1% and has not been depicted

in fig.3.) Finally, fig.4 shows the calculated aveгage electгon eneгgy Uk as а

function of E/N. ..., ....,

g

а ,..., ..., u а '-... 100 80 60 40 20 Аг CF4

U; = 15.7; Ue= 11.5 U1 = 15; Ue= 12; Uv= 0. 1 Qa = о

.

Qv= О Qa = 7 10-23 fог 5.5 S:uS: 8.5 Q; = 1 . 2 10-2 1 ( u-u i ) fог UlU; Q; = 10 -21( u-u1 ) fог uгu1

Qe = 1O-21(u-ue) fог игие Qe = 1O-21(u-ue) fог UlUe

Qem see [4] Qv = 10-20 for O.lS:uS:1 Qem= 10-19~l+u/8) fог u(8

2 10- 9 fог uг8

TabLe 1. NwnericaL data. The energies и, UJ are in units of

eV, and the cross sections QJ are in units of m2 •

In the energy range not mentioned QJ=O.

Е --- CF 4 -Аг

-10 20 30 40 50 60 70 E/N (х 0.31 1O-16Vcm2)

-

> _Q.) ___, .>: ;j 10 Uk 5 l о 4 = 5ju312f(u)du о -Аг ---· CF ₄ 8 12 16 20 E/N (х 1O-16_Vcm2 ) 24

Fig.З. Energy Loss channeLs: E-totaL ехс.,

l-totaL ion., D-diffusion, A-attachment.

Fig.4. ELectron characteristic energy as а function of EIN.

SНЕАТН PROPERTIES

28

The ion kinetics in the space charge sheaths goveгn the ion bomЪaгdment

of suгfaces and consequently play an impoгtant гоlе in plasma etching. In the

еагliег mentioned frequency гegime, the ions аге too ineгt to follow the fast potential vaгiations. This means, in effect, that they are only accelaгated in the sheath towaгds the electгodes Ьу the time aveгaged value of the sheath potential

2тт

<V(x,t)>t=

f

V(x,t)dыt/2~. (10)

о

where V(x,t) is the potential at the position х in the sheath, at time t. The influence of the instanteneous sheath potential on the ion dynamics has been discussed in [5]. In огdег to calculate the potential dгор acгoss the sheath, as well as the electгon and ion density pгofiles in the sheath, we must solve

(6)

d2<V(x,t)>t __ !:_ [ ·( ) _ < ( t)>]

d Х 2 - Е.о п. Х пе х, t {11)

Неге, ni (х) is the time independent ion density, and <пе(х, t)>t is the time

averaged electron densi ty at the posi tion х in the sheath, see fig.5. То

proceed with further calculations we make the following assumptions:

- Without any restrictions for the validity of the model we will assume that the sheath reaches its ma.ximum position as wt

= v/2, i.e. S{v/2)

= Smax, and

as а consequence of the symmetry, its minimum position as wt = Зv/2, i.e.

S(Зv/2) = Smin•

- The sheath thickness in time, S(t), which oscillates between Sm1n and Smax, is even with respect to wt = 1112, i.e. 8(11/2-Лwt) = S(11/2+Лwt), see fig.6.

- Since the applied voltage varies harmonically in time, we assume that the plasma current density can Ье approximated Ьу jp(wt) = jocos(wt), where jo is an amplitude factor.

Considering а collisionless sheath, the relation between the ion density and the time averaged electron density can Ье written as

п,(х) _ [ Vв

]½

ns - Vв- <V(x, t)>t • (12)

where Vв represents the Bohm potential, i.e. Vв = uk/e, and ns is the ion density at the plasma-sheath boundary, i.e. n5

=

n1(Smax). Неге, we assume

that the plasma ions enter the space charge sheath mono-energetically. Furthermore, we assume that the electron densi ty profile behaves step-like [6], see fig.5. This means ne(x,t)

=

п₁(х) for х > S(t), and ne(x,t)

=

О for

х ~ S(t). The time averaged electron_density profile can now Ье written as (13)

Неге, t0(x) satisfies t0(x)

= 31112

for х Е [0,Sm1n], S(to)

=

х for х Е

[Smin,Smax], and in order to obtain an unambiguous solution, t0 Е [11/2,311/2]. Substi tuting Eqs. ( 12) and ( 13) into the time averaged Poisson equation (11).

::ri _, •,-) сп с C\J "а о

s .

mtn 1 1 1n(x,t) 1 е S(t) position х

:i

·сп 'C:S ,_, .Р., 1

s

_max

Fig.5. Densities as а function of the position in the sheath.

,... ~

-а ел 10 3 eV 600 8 ~ Uk = nslni {О} = 10 о 6

~х

400 _ . ((Vsmax,t)- V(0,t}>t 4

"1'•-'

200

~о

2 Smln Smln -о 1015 ; 1016 1017 ns (m-3 }

Fig.7. S0 and the totaL potentiaL drop across the sheath as а function of ns.

А ,... < 1/1 3 111 х

,...

-

1 < ,... о

,...

-

V

-с: о

,...,

,...

(7)

we obtain dV dx =

_[

J

x-½

]1/2

½ _1✓ V [dg/dy]dy -2 [o(l-g)]

v74

1+ Smax-½ • V (1-g) - ½ where У(х) = Ув- <V(x,t)>t. 0 = en5Vв/t0, and g(x) = [3/2-tо/тт]. (14)

In order to obtain analytical expressions for the solution of equation (14). we approxirnate the integral Ьу the value -(V8/V(x))

1

/2 • Using this approxirnation. we obtain an implicit relation for the sheath potential:

Х € [О, Sm I n ] :

уз+ ЗУ~ у=

-

~½[х

-

[

1

S('rrт

2

_+тт/2)dт]

_, ₍₁₅₎

уз+ ЗУ~ у=

-

~½[х[t

0 /тт-1/2]½

-

[ФS(,rт

2

+тт/2)dт]

(16) - ½ -½½ ½ where у = [У(х) - Vв] , and Ф = (t0/тт -1/2) .

Using the mentioned approximations. and using the condi tion that the plasrna current equals the displacement current in the sheath [7], i .е. jp(t)/e

=

n1{S)dS/dt, we solved Eqs. (15) and (16). Fig.7 shows the time averaged sheath thickness S0 and the total potential drop across the sheath. and fig.8 shows the calculated electron and ion density profiles. together with the experimentally determined profile [З].

(/)

s

;:," ~ тах ~ _,... ;:," ,... ;:," ,.., n

5

~ х (/) (/)

s .

ел

-mн1 ~

-о 1Т t time т 0 27Т ; (/) ~ ,..., ..., ,..., (/) ,.., .... ~ 'tj 'tj ~ N ,..., ,-.J ~ 1... о ё: l.O ЧVsmax,t)-V(0,t)>1.= 0.8 0.6 0.4 0.2 о 2 4 6 posit ion (mm} 8 Fig.6. S(т) as а function of

time т. Dotted tine: х

>

S(т). _{and the}Fig.8. The catcutated density _{experi.тentaL}_resutts_[З].pгofites

[1] Р.М. Yallinga, et. al., 7-th ISPC. 1985. Eindhoven, The Netherlands.

[2] С.Н. Muller, and А.У. Phelps, J. Appl. Phys., 51, 6141(1980).

[З] Т.Н. Bisschops. Ph.D. Thesis. Eindhoven Univ. of Techn., 1987.

[4] Н.В. Milloy, et. al., Austr. J. Phys., 30, 61(1977). [5] Т.Н. Bisschops, et.al., 8-th ISPC, 1987, Tokyo, Japan.

[6] У.А. Godyak, and Z.Кh. Ganna, Sov. J. Plasma Phys., §, 372(1983).