Modeling of low-pressure CVD processes
Citation for published version (APA):
Kuiper, A. E. T., Brekel, van den, C. J. H., Groot, de, J., & Veltkamp, G. W. (1982). Modeling of low-pressure
CVD processes. Journal of the Electrochemical Society, 129(10), 2288-2291. https://doi.org/10.1149/1.2123495
DOI:
10.1149/1.2123495
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Published: 01/01/1982
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A. E. T. Kuiper, C. J. H. van den Brekel, J. de Groot, and G. W . Veltkamp Philips Research Laboratories, 5600 MD, Eindhoven, The Netherlands
A B S T R A C T
The spread ~n layer thickness within a series of wafers simultaneously covered in an LPCVD process will, in general, have two distinct causes. First, a spread across a wafer may occur if the deposition process is carried out in a diffusion controlled growth regime. Second, a gradual depletion in the flow direction may cause a spread within the length of the boat carrying the wafers. The latter p h e n o m e n o n can be approximated with a mathematical model. This approach reveals that the thickness spread within a batch will be acceptably small if the gas flow velocity exceeds a certain value, determined by the batch and wafer size, and also by the apparent order of the kinetics of the LPCVD process.
The a b i l i t y to grow u n i f o r m films on a large n u m b e r of substrates i n a single r u n is m a k i n g the t e c h n i q u e of low pressure CVD i n c r e a s i n g l y more p o p u l a r i n IC technology. I n the field of microelectronics silicon nitride, silicon oxide, a n d polysilicon layers w i t h t h i c k - ness v a r i a t i o n s of less than, typically, 5% for batches of over one h u n d r e d wafers of 100 m m d i a m are n o w c o m m o n l y g r o w n i n LPCVD reactors. T h e total t h i c k - ness spread in a batch will be the r e s u l t of two factors; the v a r i a t i o n of the l a y e r thickness across one w a f e r and, as a consequence of t h e i r different positions i n the reactor, the differences b e t w e e n the m e a n film thicknesses o n each wafer.
A u n i f o r m coverage of a w a f e r c a n be achieved b y choosing process conditions such that, n e a r the s u b - strate, the diffusion process b y which the reactive com- pounds are t r a n s p o r t e d to the growth surface proceeds m u c h faster t h a n the m a t e r i a l c o n s u m p t i o n b y the ac- t u a l growth reaction. I n this case, the deposition pro- cess is surface controlled. It has b e e n d e m o n s t r a t e d (1) t h a t the state of first-order process is t h e n charac- terized b y v e r y s m a l l values of the Sherwood n u m b e r
S h -- k d / D
w h e r e k is the mass t r a n s f e r coefficient of the reaction, d a r e l e v a n t l e n g t h of the order of t h e wafer spacing, a n d D the diffusion coefficient of the reactive species i n the low pressure ambient. W h e n Sh >7> 1, the deposition process is diffusion limited, w h e r e a s for Sh < < 1 the g r o w t h rate is d e t e r m i n e d b y the surface r e - action. The gain achieved b y r e d u c i n g the pressure i n a CVD reactor f r o m 1 a t m to a few t e n t h s of a Torr results from the corresponding increase of the diffusion coefficient by three orders of magnitude (see Fig. i). As a consequence, gas phase diffusion ceases to be rate determining for most reaction systems, and hence local variations in the gas phase concentration due to the geometry in the reactor will be negligibly small. Under these circumstances the growth rate is governed by the surface reaction, and homogeneously heated sub- strates (even irregularly shaped objects) will be uni- formly covered.
A sufficiently small S h number, however, in general
will not g u a r a n t e e that all wafers i n a b a t c h will be covered w i t h a l a y e r of the same thickness. A g r a d u a l depletion of the gas phase i n the flow direction will often cause a t a p e r e d thickness profile over the batch. I n this paper a t t e n t i o n is focused on a q u a n t i t a t i v e u n - d e r s t a n d i n g of this effect i n order to m i n i m i z e the o v e r a l l thickness spread o b t a i n e d i n LPCVD.
Experimental and Results
T h e low p r e s s u r e CVD of polysilicon from silane was selected as a vehicle for the p r e s e n t investigation. The reactor was a c o m m e r c i a l l y a v a i l a b l e type, viz., a n LPCVD I reactor from Applied Materials Incorporated, fitted w i t h a Roots blower. The e x p e r i m e n t a l condi- tions have b e e n described i n detail p r e v i o u s l y (1).
Key words: polysilicon, kinetics, thickness spread.
It should be n o t e d t h a t a flat t e m p e r a t u r e profile w a s
created i n the reactor, such that the t e m p e r a t u r e over the l e n g t h of the boat was constant at 625~ to w i t h i n loC.
F i g u r e 2 shows typical v a r i a t i o n s i n the observed g r o w t h rate w i t h the wafer position, w i t h the i n p u t c o n c e n t r a t i o n of silane as p a r a m e t e r . At low i n p u t c o n c e n t r a t i o n s a strong d e p l e t i o n is found, w h i c h g r a d - u a l l y diminishes w i t h i n c r e a s i n g p a r t i a l silane pres- sures. The g r o w t h rate profiles tend to flatten, u n t i l the silane gas phase c o n c e n t r a t i o n n e a r the f r o n t e n d b e - comes so high that homogeneous gas phase reactions lead to a n increased deposition on the first wafers. This effect is a c c o m p a n i e d b y a n o n u n i f o r m deposition o n the wafers, as has b e e n established i n a previous study (1). F o r the p r e s e n t i n v e s t i g a t i o n the e x p e r i m e n t a l conditions were chosen such t h a t this p h e n o m e n o n could n o t occur. F i g u r e 2 also shows a r e t a r d e d g r o w t h rate at the first five wafers, w h i c h is a t t r i b u t e d to the fact that the gas at this position' is not y e t completely at reactor t e m p e r a t u r e . To compensate for this effect we have used estimated values of G ( O ) (the g r o w t h rate at the first w a f e r ) b y e x t r a p o l a t i n g the g r o w t h r a t e profiles to z = 0.
Since the LPCVD reactor used was equipped w i t h a Roots b l o w e r whose p u m p i n g speed couId be adjusted, it was possible to p e r f o r m g r o w t h e x p e r i m e n t s at dif- f e r e n t gas flows b u t at the same reactor pressure. It appears t h a t the gas flow also plays a n i m p o r t a n t role i n the process, as is d e m o n s t r a t e d i n Fig. 3. I n this figure e x t r a p o l a t e d G (O) - v a l u e s are plotted vs. the gas flow, for various p a r t i a l silane i n p u t pressures. This
In 6
l
P=O.5 tortP=760
t
o
~
~ -.,,.
cvo=sh=1 \
Fig. 1. General growth rate curve of a CVD process. The situa- tion shown is for a process operating at atmospheric and at reduced pressures.
Vol. 129, No. I0 C V D P R O C E S S E S 2289 200 150 G(A/min)
50
~ 1 7 6
~.,~_.,
,
~
~
~
, I 10 20 30 /-,0 ~ a50
,~ z(cm)
Fig. 2. Measured growth rate profiles of polysilicon for various silane/nitrogen input ratios. The total gas flow is 400 cm3/min (STP), the pressure in the reactor is 0.5 Torr. Wafer spacing is 1 cm. PSiH4 ~-- 2 0 0 ( ~ ) , 1 5 0 ( X ) , 1 0 0 ( O ) , 7 5 ( V ) , 5 0 ( A ) , and 25 mTorr([-/).
shows t h a t a m i n i m u m gas flow is r e q u i r e d to e n s u r e t h a t at z ---- 0 the a d j u s t e d i n p u t c o n c e n t r a t i o n can b e maintained. It will b e clear that in o u r case u n a m b i g u - ous conclusions m a y be d r a w n only w h e n gas flows of at least 400 c m 3 / m i n are applied.
T o describe the influence of the depletion of the gas p h a s e o n the g r o w t h rate profile, as s h o w n in Fig. 2, it is necessary to u n d e r s t a n d the kinetics of the deposi- tion process. T o this e n d extrapolated G ( O ) values, o b t a i n e d at various silane input pressures a n d gas flows, h a v e b e e n collected in Fig. 4. W e a c h i e v e d the best fit for the points in Fig. 4 (solid curve) a p p l y i n g a relation of the k i n d A PsiH4 V2 F ( p ) :_ [1] 1 + B PSiH4V 2 This can be u n d e r s t o o d f r o m a s i m p l e m o d e l of t h e deposition of (poly)silicon. F o l l o w i n g t h e a p p r o a c h of Claassen et al. (2), the f o l l o w i n g reactions are con-
sidered:
1. Dissociation of Sill4 in the gas phase
PS~H=
150 mtorr
200
f ~ x ~ x ~ x
150
50
25
t i o 500 1000 Otot (ml/minl ..Fig. 3. Growth rate at the first wafer (extrapolated) as a func- tion of the gas flow for different silane/nitrogen input ratios; the pressure is 0.5 Torr. 3| I 20( t 9
f
'050 '100 '.150 '200 '250 Ps,.~ (tort}Fig. 4. Growth rate at the first wafer (extrapolated) as a func- tion of silane partial pressure, obtained at various gas flows and input ratios (I-l: p = 0.75 Torr, Q = 1000 cm3/min. • p = 0.50 Torr, Q = 1000 cm3/min. O : P = 0.50 Tort, Q ~ 400 cmZ/ min. A : p = 0.10 Torr, Q ~- 400 cm3/min. A : p = 0.50 Torr, Q - - 1000 cm3/min.
Sill4 ~ Sill2 + Ha having equilibrium constant KI.
2. Adsorption of Sill2
Sill2 + * ~+~-- SiHf*
with equilibrium constant /(2. The asterisks denote a free surface site.
3. Reaction of Sill2* to produce a lattice Si-atom kr
Sill2* ) Si + H2 + * where kr is the reaction rate constant. The reaction rate of 3 can now be written as
Vr = kr " [SiH~*] = kr " K~ 9 [*] 9 Psm~ [2] and [*] can be e s t i m a t e d using
[*] ---- 1 -- [SiHf*] ~- 1 - - / { 2 [ * ] PSiH2 [3] To e x p r e s s Psis2 in t h e i n p u t p a r t i a l p r e s s u r e PSiH4 w e i n t r o d u c e the d e g r e e of dissociation of the gaseous Sill4, t n e r e m r e PsiH2 ~- c~PSiH4, w h i c h u p o n i n s e r t i o n in 3 yields 1
[,]
= [ 4 ] 1 + K2 ~Psm4 and w i t h Eq. [2] w e o b t a i n kr K2 aPsiH4 Vr = [5] 1 + K2 aPsitt4 can be e x p r e s s e d in K1 using K1 = 'a2pSiH4/ (1 -- a) [6] w h i c h for s m a l l v a l u e s of a ( P s i 4 not too s m a l l ) r e - duces to a _~ (Kt/PsiH4)'/2. The r e a c t i o n r a t e LS] thus becomes kr K2K1 '/2 PSiH4 V2 v2 : [7] 1 + K~K1 '/2 PSiH4 F2 w h i c h corresponds to t h e o b s e r v e d g r o w t h kinetics a s e x p r e s s e d in Eq. [1]. It is r e m a r k e d t h a t w h e n h y d r o g e n is used as c a r r i e r gas e x p r e s s i o n [6] should r e a d aPSiH4 " PH2 K~ ---- (1 --=)PsiH4 [8] hence kl K1K2 PSiH4 v2 = [9] K1 "t- PH~ + K1K2 PSiH4 F i g u r e 4 shows t h a t at silane i n p u t p r e s s u r e s of 0.2 T o r r a n d h i g h e r r e l a t i o n [7] ceases to m a t c h the e x - p e r i m e n t a l results. As m e n t i o n e d before, Van d e nB r e k e l and B o l l e n (1) h a v e p r o v e d t h a t at such h i g h i n p u t c o n c e n t r a t i o n s gas phase n u c l e a t i o n occurs, l e a d - ing to an i n c r e a s e d d e p o s i t i o n r a t e at the first w a f e r s a n d to a d i f f e r e n t set of reactions.
The m e a s u r e d k i n e t i c s of the g r o w t h process c a n now e x p l a i n the differing shapes of t h e g r o w t h r a t e profiles, as in Fig. 2. A t l o w i n p u t c o n c e n t r a t i o n s t h e s y s t e m is m u c h m o r e sensitive to d e p l e t i o n t h a n at h i g h e r i n p u t s of r e a c t i v e species d g l / d p l > d g J d p 2 for Pl ~ P2. This is s h o w n in a s l i g h t l y d i f f e r e n t m a n n e r in Fig. 5, w h e r e the g r o w t h rate, r e l a t i v e to t h e g r o w t h r a t e at the first w a f e r (z = 0), is p l o t t e d as a f u n c t i o n of the l o n g i t u d i n a l position in t h e r e a c t o r : c o m p a r e c u r v e f~ w i t h c u r v e x.
T h e g r o w t h r a t e profile is also d e p e n d e n t on t h e gas flow, as is d e m o n s t r a t e d b y t h e u p p e r t h r e e c u r v e s in Fig. 5, o b t a i n e d at equal i n p u t c o n c e n t r a t i o n s of silane. The d e c r e a s e of the g r o w t h r a t e w i t h z is seen to d i - m i n i s h w i t h i n c r e a s i n g gas flow velocity. This influence can be u n d e r s t o o d on the basis of the d i m e n s i o n l e s s n u m b e r , the s o - c a l l e d Pdclet n u m b e r
P e = v d / D
w h e r e v is the flow velocity. I t w i l l be a p p r e c i a t e d t h a t the gas phase c o n c e n t r a t i o n w i l l be n e a r l y c o n s t a n t along t h e l e n g t h of the r e a c t o r if, e v e n at the l a s t wafer, the s u p p l y of the r e a c t i v e c o m p o u n d v i a the gas flow is l a r g e c o m p a r e d w i t h the c o n s u m p t i o n t a k i n g p l a c e at all r e a c t i v e surfaces.
F o r a b e t t e r u n d e r s t a n d i n g of the process a m a t h e - m a t i c a l m o d e l has b e e n d e v e l o p e d in o r d e r to c a l c u l a t e the r e a c t i v e species c o n c e n t r a t i o n a n d the g r o w t h r a t e as a f u n c t i o n of z.
M a t h e m a t i c a l M o d e l
W e w i l l discuss now a s i m p l e m a t h e m a t i c a l m o d e l t h a t d e s c r i b e s the g r o w t h r a t e profiles for v a r i o u s e x - p e r i m e n t a l conditions. I n the model, see Fig. 6, it is a s s u m e d t h a t the gas fiow is p a r a l l e l to t h e axis of t h e reactor, t h a t i t has a c o n s t a n t v e l o c i t y v in the flow r e g i o n (b ~ r ~ a, w h e r e a = r e a c t o r r a d i u s and b = w a f e r r a d i u s ) , a n d t h a t its v e l o c i t y is e q u a l to zero b e t w e e n the wafers. This a s s u m p t i o n is justified u n d e r the conditions n o r m a l l y p r e s e n t in L P C V D reactors, as has b e e n discussed (1). I n the s t a t i o n a r y s i t u a t i o n t h a t w e a r e c o n s i d e r i n g h e r e the r e a c t i v e gas c o n c e n t r a t i o n C ( r e l a t i v e to an e q u i l i b r i u m v a l u e ) m u s t s a t i s f y a c o n v e c t i o n - d i f f u s i o n e q u a t i o n v 0C ~C - - - - 0 , b < r < a [10a] D Oz
6{zJ/6(O)
l 10
05
~ o ~
b z l c m }Fig. 5. Normaiized growth rates as a function of z, measured at different values of Pe and Pc. These values have been estimated using D = 0.1 (T/300) 2 9 760/p cm2/sec. From Fig. 4 a value for k of the order of ! cm/sec is derived, when G(0) is exl~ressed in at/sec/cm 2.
PSiH4
Pt
(Torr) (Torr)
v(cm/sec) D(cm2sec -I)Pe
X : 0.1 0.03 1700 6850 0.25 0 : 0.5 0.03 850 1370 0.62 A : 0.5 0.03 350 1370 0.25 [ ] : 0.1 0.005 1700 6850 0.25 I ~ J I "<~:-~ ~ - / I ~
~
2~bd~2"Eb2
Fig. 6, Schematic view of the LPCVD reactor. V is the gas flow velocity, a the radius of the tube, b the wafer radius, and d the slice stacking distance.
w h e r e
1 3 O 0 ~
r Or Or Oz 2
is t h e L a p l a c e o p e r a t o r in c y l i n d r i c a l c o o r d i n a t e s a n d D t h e diffusion coefficient of t h e r e a c t i v e species. F o r 0 ~ r ~ b t h e convective t e r m in [10a] d r o p s out a n d t h e n w e h a v e for C L a p l a c e ' s e q u a t i o n
0C = 0 [10b]
At the reactor wall and at the wafer surfaces t h e
boundary co.ndition is
D --0C _ F ( C ) [11]
On
H e r e it is u n d e r s t o o d t h a t O/On r e p r e s e n t s t h e d e r i v a - tive along t h e n o r m a l v e c t o r in t h e d i r e c t i o n of the i n t e r i o r of the reactor, F is a n e x p r e s s i o n for t h e g r o w t h r a t e w i t h p r o p e r t y F (0) = 0.
A n a n a l y t i c a l solution of the p r o b l e m can b e o b - t a i n e d w h e n F ( C ) is l i n e a r in C, i m p l y i n g first o r d e r k i n e t i c s (3). H o w e v e r , for the d e p o s i t i o n r e a c t i o n of p o l y s i l i c o n this a p p e a r s not to be the case (see Eq. [1]). We t h e r e f o r e d i r e c t a t t e n t i o n to Eq. [10a] a n d [10b] and t r y to d e t e r m i n e a simplified solution. Since t h e process is in t h e r e a c t i o n l i m i t e d mode, one m a y a s s u m e t h a t the c o n c e n t r a t i o n in t h e gas p h a s e w i l l h a r d l y v a r y along t h e r a d i a l direction. W e n o w define
S : r C ( r , z) dr u ( z ) -- 2 [12] a s -- b2 F r o m Eq. [10a] w e t h e n d e r i v e a d i f f e r e n t i a l e q u a t i o n for u dzS D d z
z [
i
- - - - 0 [ 1 3 ] + a 2 _ b 2 a ~ r=a Or ~=bBy a p p l i c a t i o n of Gauss' t h e o r e m to Eq. [10b] w e find, to a first a p p r o x i m a t i o n .8C ] 2~b 2 9 .. = ~ F ( C ( b , z ) ) [14] D ar r=b 2~bd F o r r e a c t i o n - c o n t r o l l e d conditions w e m a y f u r t h e r as- s u m e a p p r o x i m a t e l y C ( a , z ) = C ( b , z ) = u ( z ) [15] By w r i t i n g ~ = z / d , a n d r e a l i z i n g t h a t w = uCo, w h e r e Co --- u ( 0 ) , w e finally a r r i v e a t the f o l l o w i n g d i f f e r - e n t i a l e q u a t i o n for w d~w d w d ad + b 2 F(Co) F ( w C o ) 0~ 2 Pe d ~ -- z = 0 D aS-- b~ CoF(Co) [16] d~u v d u
Vol. 129, No. 10 CVD P R O C E S S E S 2291
W e h a v e c o m p u t e d s o l u t i o n s of Eq. [16] n u m e r i c a l l y for v a r i o u s v a l u e s of Pe a n d Co. The r e l a t i v e g r o w t h r a t e s ( G ( z ) / G ( O ) ) o b t a i n e d f r o m this h a v e b e e n
p l o t t e d in Fig. 7. B y i n t r o d u c i n g a n e w i n d e p e n d e n t v a r i a b l e ~1, defined b y
~] = p~ [17]
w h e r e p is a constant, Eq. [16] is c o n v e r t e d into
d2w dw F (wCo) p 2 _ _ _ p p e - Q - - - 0 [18] d~] 2 du F (Co) w h e r e d a d + b 2 F(C0) Q = 2 D a 2 -- b~ Co
The scale f a c t o r p is now chosen such t h a t p = Q/Pe.
It is k n o w n (4) t h a t for s m a l l p the first t e r m of Eq. [18] vanishes e x c e p t for the v e r y end of the r e a c t o r tube, w h i c h m e a n s t h a t Eq. [18] m a y be simplified to
dw F(wCo)
- - ~ - - - = 0 [ 1 9 ]
d~ F (C0)
This r e s u l t shows t h a t w, a p a r t f r o m Co, o n l y d e p e n d s on p, w h i c h m e a n s t h a t the d i f f u s i v i t y does not p l a y a n y role in w. This was e x p e r i m e n t a l l y verified b y t h e two u p p e r c u r v e s in Fig. 5. A f u r t h e r e x a m i n a t i o n of Eq. [16] r e v e a l s t h a t t h e r e exists a u n i q u e solution w ( O w i t h w ( 0 ) --- 1 t h a t s t r i c t l y d e c r e a s e s for 0 < ~ < ~0 ( g r a d u a l d e p l e t i o n ) , w h e r e a s w ( O ~ 0 for ~ ~ ~0 ( f u l l d e p l e t i o n ) . Discussion
The g r o w t h r a t e profiles as c a l c u l a t e d for d i f f e r e n t v a l u e s of Pe, D, a n d PsiH~ s h o w t h a t t h e s p r e a d in l a y e r t h i c k n e s s w i t h i n a b a t c h r e d u c e s w i t h i n c r e a s i n g P e a n d PsiH4. The r e l a t i v e l y s m a l l g r a d i e n t s t h a t a r e c a u s e d b y the d e p l e t i o n of the gas p h a s e can t h e r e f o r e b e n e u t r a l i z e d e a s i l y b y i n c r e a s i n g the flow velocity. If t h e c a p a c i t y of the v a c u u m p u m p does not a l l o w an i n c r e a s e in t h e gas flow, the effect of a s m a l l d e p l e - t i o n m a y , in p r a c t i c e , r e a d i l y b e n e u t r a l i z e d b y c r e a t - ing a p r o p e r t e m p e r a t u r e g r a d i e n t in the L P C V D r e - actor.
I
?
o t o ! 11.0 - aZ 0.5 Psi=25 m t o r r t D "=1600 cruZ/see ~ - 0 . 1 / = i i i , , 0 10 20 30 40 5 0 z Icml =r
_o o 05 Pe =0.3 D :8000 cm21sec. P.~H1.{mI~ ) --30 ~ 2 oFig. 7. Calculated relative growth rates far various values of Pe and PSiH4. Total pressure is 0.43 Torr for (a, top) and 0.09 Torr for (b, bottom). C o m p a r i s o n of Fig. 5 and 7 r e v e a l s t h a t t h e r e is a f a i r a g r e e m e n t b e t w e e n m e a s u r e m e n t s a n d s i m u l a - tions. M e a s u r e d a n d c o m p u t e d g r o w t h r a t e profiles c o r r e s p o n d i n g to c o m p a r a b l e d e g r e e s of d e p l e t i o n at z ~_ 50 a r e f o u n d to h a v e v i r t u a l l y the s a m e c o m b i n a - tion of Pe and PSiH4.
The r e s u l t s show t h a t h i g h flow r a t e s f a v o r u n i f o r m g r o w t h r a t e profiles, the effect is m o r e effective the h i g h e r PSiH4 is. This can be a p p r e c i a t e d f r o m t h e d e - creasing a p p a r e n t o r d e r of the r e a c t i o n for h i g h e r i n p u t values, as s h o w n in Fig. 4. This is also e x p r e s s e d b y the s i m u l t a n e o u s o c c u r r e n c e of b o t h the gas flow v e l o c - i t y a n d the i n p u t c o n c e n t r a t i o n in the d e n o m i n a t o r of the f a c t o r p.
I t should be noted t h a t the c a l c u l a t e d c u r v e s for v e r y low i n p u t v a l u e s d e v i a t e f r o m t h e m e a s u r e d curves. This is due to t h e use of Eq. [1], w h i c h is less v a l i d in the low i n p u t range, w h e r e the a p p r o x i m a t i o n w i t h r e g a r d to a is no l o n g e r justified. In p r a c t i c e it was f o u n d t h a t the o r d e r of the surface r e a c t i o n shifts to u n i t y for s m a l l i n p u t values, w h i l e the e m p l o y e d m a t h e m a t i c a l r e l a t i o n , Eq. [1], t h e n counts w i t h a h a l f order. C h a r l i e r (3), who s o l v e d t h e p r o b l e m for first- o r d e r kinetics, s h o w e d t h a t in t h a t case the s h a p e of the g r o w t h r a t e profiles is no l o n g e r c o n v e x (as in Fig. 5), b u t concave, w i t h the l a z g e r p a r t of the d e p l e t i o n t a k i n g p l a c e in the first h a l f of t h e reactor. This e x - p l a i n s w h y the last p a r t of the e x p e r i m e n t a l c u r v e s for s m a l l i n p u t values, w h e r e the local c o n c e n t r a t i o n is d e c r e a s e d so f a r t h a t the r e a c t i o n a p p r o x i m a t e s first o r d e r , shows a s t r o n g d i s c r e p a n c y w i t h the c a l c u l a t e d value, b e c a u s e the d e p l e t i o n is u n d e r e s t i m a t e d in t h a t case.
The m e a s u r e d profiles (Fig. 5) h a v e a s l i g h t l y differ- ent s h a p e c o m p a r e d to the c a l c u l a t e d ones. F o r z > 35 cm the a c t u a l g r o w t h r a t e a p p e a r s to be s o m e w h a t s m a l l e r t h a n in t h e c o m p u t e d profiles. This effect is a t t r i b u t e d to t h e a b r u p t l y c h a n g i n g flow conditions at the end of the r o w of slices. This was d e m o n s t r a t e d b y e x p e r i m e n t s w h e r e a l o n g a n d a s h o r t r o w of w a f e r s w e r e coated, ( b o a t filled up to z : 75 and 25 cm, r e s p e c t i v e l y ) all o t h e r conditions b e i n g k e p t con- stant. T h e g r o w t h r a t e at position z = 25 cm, was f o u n d to be 10-15% h i g h e r for the l a r g e batch. F o r the c a l - culations a n infinitely l o n g t u b e a n d r o w of w a f e r s is assumed. C o n s e q u e n t l y this effect is not t a k e n into account in the model.
S u m m a r i z i n g , one m a y s t a t e t h a t using the m o d e l as d e s c r i b e d in this p a p e r , it has b e e n p o s s i b l e to e s t a b - lish c l e a r l y t h e influence of c o m m o n L P C V D p a r a m e - ters l i k e gas flow, i n p u t p a r t i a l p r e s s u r e , t o t a l p r e s s u r e , a n d the kinetics of the d e p o s i t i o n process on t h e g r o w t h r a t e profile. T h e r e f o r e , o p t i m i z a t i o n of L P C V D processes t h a t a r e a p p l i e d in IC t e c h n o l o g y has n o w b e c o m e easier.
M a n u s c r i p t s u b m i t t e d Nov. 10, 1981; r e v i s e d m a n u - s c r i p t r e c e i v e d M a r c h 8, 1982.
A n y discussion of this p a p e r w i l l a p p e a r in a Discus- s i o n S e c t i o n to b e p u b l i s h e d in t h e J u n e 1983 JOURNAL. A l l discussions for the J u n e 1983 Discussion S e c t i o n should be s u b m i t t e d b y Feb. 1, 1983.
Pubtication costs of this article were assisted by Philips Research Laboratories.
R E F E R E N C E S
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3. J.-P. C h a r l i e r , IEEE Trans. Electron Devices, ed-28,
501 (1981).
4. R. E. O ' M a l l e y Jr., "In~troduction to S i n g u l a r P e r t u r - bations," A c a d e m i c Press, New Y o r k (1974).
