The kinetics of silicon dioxide chemical vapour deposition II: The model of the process

Surface Technology, 25 (1985) 315 - 325

315

THE KINETICS OF SILICON DIOXIDE CHEMICAL VAPOUR DEPOSITION II: THE MODEL OF T H E PROCESS PIOTR B. GRABIEC Institute o f Electron Technology CEMI, Al. Lotnik6w 32/46, 02-668 Warsaw (Poland)

JAN PRZYLUSKI Institute o f General Chemistry and Inorganic Technology, Warsaw Technical University, ul. Noakowskiego 3, 00-663 Warsaw (Poland)

(Received October 10, 1984)

Summary This paper presents a general analysis of chemical vapour deposition (CVD) o f silicon dioxide by oxidizing silane with oxygen. The proposed model is based on the scheme of surface reactions described in Part I. The model considers all the i m p o r t a n t process steps, namely mass transport in the gas phase, adsorption and chemical reactions. The proposed approach makes it possible to analyse the kinetics o f the process for a wide range of parameters. However, the resulting equation is complicated and includes some u n k n o w n coefficients. In a Part III these coefficients will be determined by fitting the model to the experimental data and the experimental verification o f the model will be presented.

1. I n t r o d u c t i o n In Part I [1] the analysis of the electron structures o f reagents which take part in the chemical vapour deposition of SiO2 was presented. As a result o f these considerations a scheme of surface reactions was proposed. In this paper, the mathematical m ode l of the process, based on this scheme, is presented. 2. Modelling the process o f chemical vapour deposition (CVD) o f SiO 2 The following general assumptions have been adopted. (i) The reactor operates in a quasi-steady state. (ii) Before th e start of the process t he substrate is covered with a thin (about 2 nm) layer of native oxide, which means t hat from the very beginning the reaction proceeds on an SiO2 surface and the deposition starts w i t h o u t any induction period. 0376-4583/85/$3.30

© Elsevier Sequoia/Printed in The Netherlands

316

(iii) The components of the gas phase are thoroughly mixed in the main stream while a diffusion (stagnant) layer exists above the substrate. (iv) Because of the low process temperature surface migration of deposited products is negligibly small. (v) Chemical reaction does not proceed in the vapour phase. Since the aim of this paper is not the analysis of any specific CVD reactor, but the process itself, an element AS of the substrate wafer together with a volume of gas above it are considered. The dimensions of this element are small in comparison with the whole substrate wafer and the reactor. Therefore, it is possible to adopt the following assumptions. (vi) For all fluxes and concentrations bc

bc

-0

by

8z

bJ

-0

by

bJ

-0

8z

-0

(vii) 8T

8T

-0

by

8z

-0

(viii) 8b

bb

-0

by

bz

-0

Some of the above assumptions were adopted after Korec and Borkowicz [2], Eversteyn and coworkers [3, 4]. The characteristic regions of the system have been selected (Fig. 1) on the above assumptions and taking into consideration the results of the Part I

[11.

main gas stream

°,[5]

,.I/ l'/ T

"

/o

LO,]

adsorption

si~c21

<

• " [o211

j

t,. l

t~.

gas-solid

--

interface

I

~ "

"'".,-., [e, ]

Fig. 1. The characteristic regions in the SiO2 deposition system.

317 2.1. T h e p a r t i a l p r o c e s s e s

A c c o r d i n g to Fick's first law, t h e diffusion flux m a y be described b y OCa

(1)

J = --DA~n a --

Ox

w h e r e J is the diffusion flux (mol s-1 cm-2), DAB is t h e c o e f f i c i e n t d i f f u s i o n o f gas A into gas B (cm 2 s-l), 3 c a / 3 x is the c o n c e n t r a t i o n gradient o f A (cm 1) and na is the t o t a l c o n c e n t r a t i o n o f gas m o l e c u l e s (mol cm-3); f o r the ideal gas na = P / R T . T h e diffusion c o e f f i c i e n t is given by the semiempirical Chen and O t h m e r e q u a t i o n [5]

DAB = 1.52 × 10 2

T1"Sl(1/MA + 1/MB)0"S P(TcA TcB)°'14°5(VcA 0"4 + Vc13°-4)2

(2)

w h e r e MA, M B are t h e m o l e c u l a r masses o f A and B, respectively (g mol-1), TeA, Tea are the critical t e m p e r a t u r e s o f gases A and B, respectively (K), V~A, VcB are t h e critical v o l u m e s o f gases A and B, respectively (cm 3 mo1-1) and P is the t o t a l pressure. C o m b i n i n g eqns. (1) and (2) and substituting P / R T f o r na we o b t a i n J = --0.0152

(1/MA + 1/Ms) °'5

(TcATcB)°.14°S(VcA °.4 +

ycB°.4)2R

T0.Sl ~Ca 5x

(3)

or, a f t e r simplification, J =--WABT 0.sl ()Ca 3x

(4)

w h e r e W A B r e p r e s e n t s a characteristic c o e f f i c i e n t o f d i f f u s i o n o f gas A into gas B. Using t h e s u b s t i t u t i o n ~ca Ox

~ca OT -

(5)

OT ~x

and assuming t h e linearity o f t h e t e m p e r a t u r e profile T = Ts

Ts-- Tg

- -

b

x

(6)

we o b t a i n A T ~c~ J = - - WABT0-Sl _ _ b ~T

(7)

S e p a r a t i o n o f variables and i n t e g r a t i o n o f eqn. (7) w i t h i n t h e d i f f u s i o n (stagnant) layer limits lead t o AT J = 0-19WAB b ( T o. 19 __ TgO.19) (c A - - CAi )

(8)

318 where cA is the concentration of gas A in the main stream and CAi is the concentration of gas A in the vicinity of the surface. The above description of diffusive transport is based on the results obtained by Van der Putte et al. [6]. An analogous description was successfully adopted by Korec and Borkowicz [2]. In the process under consideration, the resultant diffusion flux is given by the superposition of the diffusion flux of the reagents directed towards the surface and the reverse diffusion flux directed towards the main stream: Jl = Jla --J12

for oxygen

J2 = J21 -- J22

for silane

(9) (10)

Using eqn. (8) we obtain

Jll = klCl

(11)

J12 = klCli

(12)

J21 = k2c2

(13)

J22 = k2c2i

(14)

where AT

k 1 = 0.19W13 b(TsO.19 - TgO.19 ) k2 = 0.19W23

AT

b(Ts 0 - 1 9 - Tg 0.19)

(15) (16)

W13 is the characteristic coefficient of diffusion of oxygen (index 1) into the inert g a s - nitrogen (index 3) and W23 stands for the characteristic coefficient of diffusion of silane (index 2) into nitrogen. Because of the low process temperature and the similarity of the molecular masses of oxygen and silane, the thermodiffusion of reagents may be disregarded. Following the Langmuir theory and adopting the absolute rate theory of heterogeneous reactions at the solid surface, the adsorption flux may be expressed as [ 7, 8 ] Jags

= lesT C m C s f* - - exp ( -- Ea*t h F~f~ RT/

(17)

where kB is the Boltzmann constant, h is Planck's constant, f* is the partition function of the activated complex after disregarding a component responsible for oscillation along the reaction coordinate, Fg is the partition function of the reagent in the gas phase, fs is the partition function of surface active centres, cm is the molar concentration of the reagent in the gas :phase, cs is the surface concentration of free active centres and Ea* is the activation energy in forming the active complex during the adsorption.

319

Similarly, the desorption flux is Jdes-

kBT h

f* Ed* C a - - exp - - - fa RT

(18)

where Ea* is the activation energy in forming the active complex during the desorption, fa is the partition function of the adsorbed molecule and Ca is the surface concentration of active centres occupied by adsorbed molecules. Unfortunately, it is very difficult to c o m p u t e the values of partition functions of active centres. However, by introducing fractions of the surface N+ O+

Ol

-

Nx

02

N~:

N_

N2

N 1

-

-

O_

N~

-

N~

(19)

01+05+0_+0+=1 adsorption and desorption fluxes may be described as follows: J13 = h l a C l i O +

(20)

J23 = k23C2i0+

(21)

J14 = k 140 1

(22)

J24 = k2402

(23)

It is assumed that nitrogen adsorption is negligibly small. Since the activation energy of chemisorption is high, the linear term of the temperature dependence is negligibly small as compared with the exponential terms, hence k13 = kl°3 exp

RT]

k:a = k°3 exp

RT]

k14 = k°4 exp

RT]

(26)

k24 = k2°4 exp

RT]

(27)

(24)

In agreement with the proposed mechanism of surface reactions the dissociation flux of silane is

J2s =

riO20-

--

~0si~30H

where k" and k are the rate constants of the reactions proceeding towards the right and left, respectively. 0sin3 and 0H are fractions of the surface covered b y Sill4 dissociation products. In view of the assumption of a high hydrogen separation rate i.e. OH ~ 0, the above relation may be smplified to give

320

J2s = k2s020k2s = h°s e x p ( -E2slRT]

(28)

Similarly, the oxygen dissocation may be described by the equation

Jls = kls010+

(__ E15t kas = k°s exp

(29)

RT]

2.2. The model o f Si02 chemical vapour deposition As a result of the above considerations, a system of equations describing the process is obtained: Jll = klCl

diffusion of oxygen to the surface

(30)

J12 = klCli

diffusion of oxygen to the main stream

(31)

J13= kl3CliO+

adsorption of oxygen

(32)

J14 = k1401

desorption of oxygen

(33)

da5 = kls010+

dissociation of oxygen

(34)

J21 = k2c2

diffusion of silane to the surface

(35)

J22 = k2c2i

diffusion of silane to the main stream

(36)

J23 = k23c2i0+

adsorption of silane

(37)

J24 = k2402

desorption of silane

(38)

J25 = k2s020-

dissociation of silane

(39)

where the dependence of the surface processes on temperature is described by the general equation

kij = k ,j9- exp 1--

RT]

fori =lor2"j, =3,4,5

(40)

and the dependence of diffusive mass transport on temperature is described by eqns. (15) and (16). The deposited layer is stoichiometric for a wide range of process parameters. This observation and the stochastic character of the surface processes (examined on an atomic scale) lead us to the supposition that

O_ = 20_,.

(41)

In the steady-state, when the generation/recombination phenomena may be disregarded, the continuity equations in particular regions are:

321

for oxygen J11 + J14 = J13 + J12

(42)

J,a = Jls +J14

(43)

for silane g21 + J24 = J23 + J22

(44)

J23 = J25 + J14

(45)

When the deposition rate f is expressed in the same units as the fluxes (i.e. mol s-1 cm -2) equations (30) - (39) may be written as follows: resultant diffusion rate of oxygen

(46)

resultant.diffusion rate of silane

(47)

k1401

resultant adsorption rate of oxygen

(48)

k240 2

resultant adsorption rate of silane

(49)

= kls010+

resultant dissociation rate of oxygen

(50)

= k2s020

resultant dissociation rate of silane

(51)

= klC

klCli

1 --

= k2C 2 - - k2c2i

f

=

kl3CliO+

--

= k23c2i0

+ --

Substituting for 0_ from eqn. (41) in eqn. (51) and comparing with eqn. (50) we obtain 01 - 2k25 02 kls

(52)

From eqns. (48) and (49) we obtain k23c2i

f + k240 2

kl3Cli

r + ~1401

(53)

Hence, introducing substitute coefficients, A, -

k15k24

kls + 2k25 A2 =

A3 -

3klSk242 2k25(klS + 2k2s ) 2k2sk 14k25

(54)

(55)

(56)

k13k24k15

A4 -

k23

(57) kl3 and taking into consideration eqns. (19), (41), (52) and (53), after some manipulation, the deposition rate may be written as

322

r= A1

(A3c2i

--Cli

)

(A3c2i

--Az

(Cli -- A4c2i)

--Cli)

2

(Cli -- A4c2i)

2

(58)

E q u a t i o n (58) describes t h e d e p o s i t i o n rate w i t h o u t considering t h e diffusive t r a n s p o r t o f reagents. If t h e d i f f u s i o n processes are to be t a k e n into a c c o u n t , t h e c o n c e n t r a t i o n o f the reagents near the surface (c~i, c2i) should be replaced by the c o n c e n t r a t i o n o f t h e reagents in the main stream. F r o m eqns. (46) and (47) we o b t a i n 1 Cli C 1 --

--

F

(59)

kl

1 C2i = C 2 - -

--

(60)

F

k2

C o m b i n i n g eqns. (59) and (60) with eqn. (58), after some m a n i p u l a t i o n we obtain (Aakl -- k2)2r 3 +

--

+

I 2(A4k I -- k2)

klk2

(c 1 -- A4c2) +

Al(k2--A3kl)(A4kl--k2)ll k12k2 2

/.24"

2A2(k2 - - A 3 k l ) -- A l(A4k i -- k2)

klk2

A2(k2 -- A3kl) 2

kaZk2 2

(C 1 -- A4c2) 2

(A3c 2 -- Cl)

A l ( k 2 -- A3kl) -- A4c2) I r A2(A3c 2 Cl) 2 klk2 (Cl + --- A I(A 3c2 -- cl)(cl

-

-

A4c2)

= 0

(61)

a f t e r c o m b i n i n g the d e p o s i t i o n rate r and reagent c o n c e n t r a t i o n s Cl and c2. T h e above e q u a t i o n is c o m p l i c a t e d and difficult to i n t e r p r e t . It should be n o t i c e d t h a t according to t h e e x p e r i m e n t a l d a t a [ 8 - 11], t h e relationship b e t w e e n t h e d e p o s i t i o n rate and c o n c e n t r a t i o n o f silane or o x y g e n does n o t suggest a t h i r d - o r d e r e q u a t i o n . R e d u c t i o n o f eqn. (61) t o t h e s e c o n d - o r d e r m a y be a c c o m p l i s h e d in t h e t w o ways: (a) assuming r3~ 0

(62)

and i n t r o d u c i n g c o e f f i c i e n t s k2--A3kl 8 1 --

(63)

klk2

323

A4k 1--k 2 $2

(64)

-

klk2 PI = A2812 - - A 1 S 1 S :

(65)

P2 = A4

(66)

P3 = ( A I A 4 + 2A2Aa)S1 - - A 1 A a S 2

(67)

P4 = (AI + 2 A 2 ) $ 1 - A1S2

(68)

Ps = A1 + A2

(69)

P6 = A a ( A I A 4 + A2A3)

(70)

P7 = A I A a + A 1 A 4 + 2 A 2 A 3

(71)

P8 = 2S2

(72)

P9 = 2A4S2

(73)

we o b t a i n a s e c o n d - o r d e r e q u a t i o n

(Pscl - - P 9 c 2 + P1)r 2 + ((C 1 - - P2C2) 2 + P3c2 - - P 4 C l } r + Pscl 2 + P6c22 --P7CLC2 = 0

(74)

(b) assuming A4kl ~ k 2

(75)

eqn. (61) m a y be r e w r i t t e n in t h e f o l l o w i n g f o r m :

P1 r2 + ((cl - - P2c2) 2 + Pac2 - - P 4 c l } r + PsCl 2 + P6c22 - - P 7 c l c 2 = 0

(76)

In this case, since $2 = 0,

PI = A2S12

(77)

Pa = ( A 1 A 4 + 2A2A3)S1

(78)

P4 = (A1 + 2A2)S1

(79)

E q u a t i o n s (63), (66) and (69) - (73) a p p l y in b o t h cases. The s o l u t i o n o f eqn. (74) o r (76) is

r~-

--

--

t(£; °f --

(80)

X

where D = PsCl 2 + P6c22 - - P7CLC2

(81)

B = (cl - - P2c2) 2 + Pac2 - - P 4 c l

(82)

and

A = P8cl - - P9c: + P1

f o r variant (a) (eqn. (74))

(83)

324 or

A = P1

f o r variant (b) (eqn. (76))

(84)

The solution r = -

+

-

(85)

h a s b e e n t e s t e d and c o u l d n o t be f i t t e d to t h e e x p e r i m e n t a l d a t a . 3. C o n c l u s i o n s It has b e e n d e m o n s t r a t e d t h a t a t h e o r e t i c a l m o d e l o f silicon d i o x i d e C V D can be c o n s t r u c t e d , c o m b i n i n g various t y p e s o f g r o w t h c o n t r o l and b a s e d o n t h e s c h e m e o f s u r f a c e r e a c t i o n s d e s c r i b e d in P a r t I. T h e d e p e n d e n c e o f t h e SiO 2 d e p o s i t i o n r a t e o n t h e process p a r a m e t e r s is d e s c r i b e d b y eqns. (80) - (84). T h e c o e f f i c i e n t s P1 . . . . , P9, w h i c h a p p e a r in t h e s e e q u a t i o n s , are d e f i n e d b y eqns. (63) - (73), (77) - (79) a n d (54) - (57). T h e d e p e n d e n c e o f t h e r a t e c o n s t a n t s i n c l u d e d in t h e c o e f f i c i e n t s P1, -.., P9 o n t h e t e m p e r a t u r e is given in eqns. {15), (16) a n d (40). T h e resulting s y s t e m o f e q u a t i o n s (eqns. (80) - (84)) is c o m p l i c a t e d a n d includes u n k n o w n coefficients P1 . . . . , P9. Since their values c a n n o t be d e f i n e d t h e o r e t i c a l l y with s u f f i c i e n t p r e c i s i o n , it is n e c e s s a r y to c o m p u t e t h e m b y fitting t h e t h e o r e t i cal f o r m u l a to t h e e x p e r i m e n t a l d a t a . This p r o b l e m , as well as t h e c o m p a rison o f t h e m o d e l w i t h e x p e r i m e n t a l d a t a and c o n c l u s i o n s resulting f r o m t h e m o d e l will be discussed in t h e Part I I I o f this s t u d y .

References 1 P. B. Grabiec and J. Przyluski, Surf. Technol., 25 (1985) 307. 2 J. Korec and J. Borkowicz, Electron Technol., 10 ( 1 9 7 7 ) 3. 3 F. Eversteyn, P. Severin and C. Van der Brekel, J. Electrochem. Soc., 117 (1970)

925. 4 F. Eversteyn, Philips Res. Rep., 29 (1974) 45. 5 R. Pohorecki and S. Wrofiski, Kinetyka i termodynamika Procesdw Inzynierii Chemicznej, WNT, Warsaw, 1979. 6 P. Van der Putte, L. G. Gilling and J. Bloem, J. Cryst. Growth, 31 (1975) 299. 7 P. Barret, Cindtique Hdtdrog~ne, Gauthier-Villars, Paris, 1973. 8 G.M. Barrow, Physical Chemistry, McGraw-Hill, New York, 1973. 9 W. Kern and G. L. Schnable, IEEE Trans. Electron Devices, 26 (1979) 647. 10 E. A. Taft, J. Electrochem. Soc., 126 (1979) 1728. 11 C. Cobianu and C. Pavelescu, J. Electrochem. Soc., 130 (1983) 1888.

Appendix A: n o m e n c l a t u r e b cl

diffusion layer thickness (cm) o x y g e n c o n c e n t r a t i o n in t h e m a i n s t r e a m ( m o l mo1-1)

325 Cli c2 c2i Eii

Jij kij k° k1 k2 N N~ N2 N÷ N_ Tg Ts x y, z

oxygen concentration in the gaseous phase in the vicinity of the substrate (mol mo1-1) silane concentration in the main stream (mol mo1-1) silane concentration in the gaseous phase in the vicinity of the substrate (mol m o l - l) activation energy for the process j required by the reagent i (J mo1-1) (i, 1 for oxygen; i, 2 for silane;j, 3 for adsorption;j, 4 for desorption; j, 5 for dissociation) the flux of reagent i in the process j (mol s-1 cm -2) (i and j are the same as for Eij) rate constant of the process j of the reagent i (mol s-1 cm -2) (i and ] are the same as for Ei/) pre-exponential factor of the rate constant hi1 (mol s-1 cm -2) mass transport coefficient describing the diffusion of oxygen (mol S- 1 cm-2) mass transport coefficient describing the diffusion of silane (mol s-1 cm -2) the total number of active sites taking part in the process (cm 2) the number of active silicon sites occupied by adsorbed oxygen molecules (cm -2) the number of active silicon sites occupied by adsorbed silane molecules (cm -2) the number of free positively charged silicon active sites (cm -2) the number of free negatively charged oxygen active sites (cm -2) deposition rate (mol s-1 cm -2) temperature of gas in the main stream (K) surface temperature of the substrate (K) coordinate perpendicular to the surface coordinates parallel to the surface