1. business and industrial
2. manufacturing

Modeling of Low-Pressure CVD Processes
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
ABSTRACT
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 ability to grow u n i f o r m films on a large n u m b e r
of substrates in a single r u n is m a k i n g the technique
of low pressure CVD increasingly more popular in IC
technology. I n the field of microelectronics silicon
nitride, silicon oxide, a n d polysilicon layers w i t h thickness variations of less than, typically, 5% for batches
of over one h u n d r e d wafers of 100 m m diam are now
commonly grown i n LPCVD reactors. The total thickness spread in a batch will be the result of two factors;
the variation of the layer thickness across one wafer
and, as a consequence of their different positions in
the reactor, the differences b e t w e e n the m e a n film
thicknesses on each wafer.
A u n i f o r m coverage of a wafer can be achieved b y
choosing process conditions such that, near the substrate, the diffusion process b y which the reactive compounds are transported to the growth surface proceeds
much faster t h a n the material consumption by the actual growth reaction. I n this case, the deposition process is surface controlled. It has been demonstrated
(1) that the state of first-order process is then characterized by very small values of the Sherwood n u m b e r
Sh -- k d / D
where k is the mass transfer coefficient of the reaction,
d a r e l e v a n t length of the order of the wafer spacing,
and D the diffusion coefficient of the reactive species
in the low pressure ambient. W h e n Sh >7> 1, the
deposition process is diffusion limited, whereas for Sh
< < 1 the growth rate is d e t e r m i n e d by the surface reaction. The gain achieved by reducing the pressure i n
a CVD reactor from 1 a t m to a few tenths 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 substrates (even irregularly shaped objects) will be uniformly covered.
A sufficientlysmall Sh number, however, in general
will not guarantee that all wafers in a batch will be
covered with a layer of the same thickness. A gradual
depletion of the gas phase i n the flow direction will
often cause a tapered thickness profile over the batch.
In this paper attention 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 in order to minimize the
overall thickness spread obtained in LPCVD.
Experimental and Results
The low pressure CVD of polysilicon from silane was
selected as a vehicle for the present investigation. The
reactor was a commercially available type, viz., an
LPCVD I reactor from Applied Materials Incorporated,
fitted with a Roots blower. The e x p e r i m e n t a l conditions have been described in detail previously (1).
Key words: polysilicon, kinetics, thickness spread.
2288
It should be noted that a flat t e m p e r a t u r e profile w a s
created in the reactor, such that the t e m p e r a t u r e over
the length of the boat was constant at 625~ to w i t h i n
loC.
Figure 2 shows typical variations in the observed
growth rate with the wafer position, with the i n p u t
concentration of silane as parameter. At low i n p u t
concentrations a strong depletion is found, which gradually diminishes with increasing partial silane pressures. The growth rate profiles tend to flatten, u n t i l the
silane gas phase concentration n e a r the front end becomes so high that homogeneous gas phase reactions
lead to a n increased deposition on the first wafers. This
effect is accompanied by a n o n u n i f o r m deposition on
the wafers, as has b e e n established i n a previous study
(1). For the present investigation the e x p e r i m e n t a l
conditions were chosen such that this p h e n o m e n o n
could not occur. Figure 2 also shows a retarded growth
rate at the first five wafers, which is a t t r i b u t e d to the
fact that the gas at this position' is not yet completely
at reactor temperature. To compensate for this effect
we have used estimated values of G ( O ) (the growth
rate at the first wafer) b y extrapolating the growth
rate profiles to z = 0.
Since the LPCVD reactor used was equipped with a
Roots blower whose p u m p i n g speed couId be adjusted,
it was possible to p e r f o r m growth experiments at different gas flows but at the same reactor pressure. It
appears that the gas flow also plays a n i m p o r t a n t role
in the process, as is demonstrated i n Fig. 3. I n this
figure extrapolated G (O) -values are plotted vs. the
gas flow, for various partial silane i n p u t pressures. This
In 6
l
P=O.5tort
P=760t
o
~
cvo=sh=1
~ -.,,.
\
Fig. 1. General growth rate curve of a CVD process. The situation shown is for a process operating at atmospheric and at reduced
pressures.
Vol. 129, No. I0
CV D P R O C E S S E S
2289
3| I
200
20(
~176
150
G(A/min)
~.,~_.,
,
~
~
t9
f
~
'050
50
,
I
10
20
30
/-,0~ a
50
,~ 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 ( ~ ) , 150(X), 1 0 0 ( O ) , 75(V), 50(A), and
25 mTorr([-/).
'100
Ps,.~ (tort}
'.150
'200
'250
Fig. 4. Growth rate at the first wafer (extrapolated) as a function 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 equilibriumconstant KI.
2. Adsorption of Sill2
Sill2 + * ~+~--SiHf*
shows that a m i n i m u m gas flow is r e q u i r e d to ensure
that at z ---- 0 the adjusted input concentration can be
maintained. It will be clear that in our case unambiguous conclusions may be drawn only when gas flows
of at least 400 cm3/min are applied.
To describe the influence of the depletion of the gas
phase on the growth rate profile, as shown in Fig. 2,
it is necessary to understand the kinetics of the deposition process. To this end extrapolated G(O)
values,
obtained at various silane input pressures and gas
flows, have been collected in Fig. 4. We achieved the
best fit for the points in Fig. 4 (solid curve) applying
a relation of the kind
A PsiH4 V2
F ( p ) :_
[1]
1 + B PSiH4V2
This can be u n d e r s t o o d f r o m a simple 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 approach of
Claassen et al. (2), the f o ll o w in g reactions are considered:
1. Dissociation of Sill4 in the gas phase
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 est i m at ed using
[*] ---- 1 -- [SiHf*] ~- 1 - - / { 2 [ * ] PSiH2
[3]
To express Psis2 in the input p ar t i al pressure PSiH4
w e i n t r o d u ce the degree of dissociation of the gaseous
Sill4, t n e r e m r e PsiH2 ~- c~PSiH4,w h i c h upon insertion in
3 yields
1
[,]
=
[4]
1 + K2 ~Psm4
and w i t h Eq. [2] we obtain
PS~H=150mtorr
200
Vr =
kr K2 aPsiH4
[5]
1 + K2 aPsitt4
f~x~x~x
can be expressed in K1 using
K1 = 'a2pSiH4/(1 -- a)
150
50
25
[6]
w h i ch for small values of a ( P s i 4 not too small) r e duces to a _~ (Kt/PsiH4)'/2. The reaction rate LS] thus
becomes
kr K2K1 '/2 PSiH4V2
v2 :
[7]
1 + K~K1 '/2 PSiH4F2
w h i ch corresponds to the observed g r o w t h kinetics a s
expressed in Eq. [1].
It is r e m a r k e d that w h e n h y d r o g e n is used as c a r r i e r
gas expression [6] should read
aPSiH4 " PH2
K~ ---- (1 --=)PsiH4
[8]
hence
t
o
500
i
1000
Otot (ml/minl
..
Fig. 3. Growth rate at the first wafer (extrapolated) as a function of the gas flow for different silane/nitrogen input ratios; the
pressure is 0.5 Torr.
v2 =
kl K1K2 PSiH4
K1 "t- PH~ + K1K2 PSiH4
[9]
F i g u r e 4 shows that at silane input pressures of 0.2
T o r r and higher relation [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 n
2290
J. E l e c t r o c h e m . Soc.: S O L I D - S T A T E S C I E N C E A N D T E C H N O L O G Y
Brekel and Bollen (1) h a v e p r o v e d that at such high
input concentrations gas phase nucleation occurs, l e a d ing to an increased deposition r a t e at the first wafers
and to a different set of reactions.
The m e a s u r e d kinetics of the g r o w t h process c a n
now e x p l a i n the differing shapes of the g r o w t h r a t e
profiles, as in Fig. 2. A t low i n p u t concentrations the
system is much m o r e sensitive to depletion t h a n at
h i g h e r inputs of reactive species dgl/dpl > dgJdp2 for
Pl ~ P2. This is s h o w n in a slightly different 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 the g r o w t h
rate at the first w a f e r (z = 0), is plotted as a function
of the l o n g i t u d i n a l position in the reactor: c o m p a r e
curve f~ with curve x.
The 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 the u p p e r t h r e e curves in
Fig. 5, obtained at equal i n p u t concentrations of silane.
The decrease of the g r o w t h rate w i t h z is seen to diminish w i t h increasing gas flow velocity. This influence
can be u n d e r s t o o d on the basis of the dimensionless
number, the s o - c a l l e d Pdclet n u m b e r
Pe = v d / D
w h e r e v is the flow velocity. It will be a p p r e c i a t e d t h a t
the gas phase concentration will be n e a r l y constant
along the l e n g t h of the r e a c t o r if, even at the last
wafer, the s u p p l y of the r e a c t i v e compound via the
gas flow is large c o m p a r e d w i t h the consumption
t a k i n g place at all reactive 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 been developed in o r d e r to calculate
the reactive species concentration and the g r o w t h r a t e
as a function of z.
Mathematical
Model
We will discuss now a simple m a t h e m a t i c a l m o d e l
that describes the g r o w t h rate profiles for various e x p e r i m e n t a l conditions. In the model, see Fig. 6, it is
assumed t h a t the gas fiow is p a r a l l e l to the axis of the
reactor, that it has a constant velocity v in the flow
region (b ~ r ~ a, w h e r e a = reactor radius and b =
wafer r a d i u s ) , and that its velocity is equal to zero
b e t w e e n the wafers. This assumption 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 LPCVD reactors, as
has been discussed (1). In the s t a t i o n a r y situation t h a t
w e are considering here the reactive gas concentration
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 satisfy a
convection-diffusion equation
~C
~I
~J
I "<~:-~ ~ -
~
2~bd~2"Eb2
/
I
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.
where
1
3
O
0~
r
Or
Or
Oz2
is the Laplace o p e r a t o r in c y l i n d r i c a l coordinates a n d
D t h e diffusion coefficient of the reactive species. F o r
0 ~ r ~ b the convective t e r m in [10a] drops out a n d
t h e n we have for C Laplace's equation
0C = 0
0C
----0,
D Oz
b<r<a
[10b]
At the reactor wall and at the wafer surfaces
boundary co.nditionis
D --0C _ F ( C )
On
the
[11]
Here 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 the d e r i v a tive along t h e n o r m a l vector in t h e direction of the
i n t e r i o r of the reactor, F is an expression for the
g r o w t h rate with 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 be 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
kinetics (3). However, for the deposition r e a c t i o n of
polysilicon 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 direct attention 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
the process is in the r e a c t i o n l i m i t e d mode, one m a y
assume t h a t the concentration in the gas phase will
h a r d l y v a r y along the r a d i a l direction. We now define
S:
r C ( r , z) dr
u ( z ) -- 2
v
O c t o b e r 1982
[12]
a s -- b2
[10a]
F r o m Eq. [10a] w e then d e r i v e a differential equation
for u
6{zJ/6(O)
l 10
~o~
05
d~u
v du
dzS
D
+
-
-
dz
z[
a2_b2
i
a ~
r=a
-
Or
~=b
-
0
[13]
By application 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
b
.8C ]
zlcm}
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
X :
0 :
A:
[]:
(Torr) (Torr) v(cm/sec) D(cm2sec -I)
0.1
0.5
0.5
0.1
0.03
0.03
0.03
0.005
1700
850
350
1700
6850
1370
1370
6850
Pe
0.25
0.62
0.25
0.25
D ar
9..
2~b 2
2~bd
= ~ F ( C ( b , z ) )
r=b
[14]
F o r r e a c t i o n - c o n t r o l l e d conditions we m a y f u r t h e r assume 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 realizing t h a t w = uCo, w h e r e
Co -----u ( 0 ) , we finally a r r i v e at the following differe n t i a l equation for w
d~w
0~2
dw
Pe
d~
-- z
d
ad + b 2 F(Co) F(wCo)
D
aS-- b~
CoF(Co)
= 0
[16]
Vol. 129, No. 10
CVD P R O C E S S E S
W e have c o m p u t e d solutions of Eq. [16] n u m e r i c a l l y
for various values of Pe and Co. The r e l a t i v e g r o w t h
rates ( G ( z ) / G ( O ) ) o b t a i n e d from this have been
p l o t t e d in Fig. 7. By 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
p2___ppe
d~]2
dw
F (wCo)
- - - 0
F (Co)
-Q
du
[18]
where
Q=2
d a d + b 2 F(C0)
D a 2 -- b~
Co
The scale factor p is now chosen such that p = Q/Pe.
It is k n o w n (4) t h a t for small 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, which means t h a t Eq. [18] m a y be simplified to
dw
- -
F(wCo)
~-
d~
-
-
= 0
[19]
F (C0)
This result shows that w, a p a r t f r o m Co, o n l y depends
on p, w h i c h means t h a t the diffusivity 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 the
two u p p e r curves 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 decreases 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 (full d e p l e t i o n ) .
Discussion
The g r o w t h r a t e profiles as calculated for different
values of Pe, D, and PsiH~ show that the s p r e a d in l a y e r
thickness w i t h i n a b a t c h reduces w i t h increasing Pe
and PsiH4. The r e l a t i v e l y small g r a d i e n t s that a r e
caused b y the depletion of the gas phase can t h e r e f o r e
be n e u t r a l i z e d easily b y increasing 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 allow
an increase in the gas flow, the effect of a small d e p l e tion may, in practice, r e a d i l y be 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 LPCVD r e actor.
I
o
?
to !
11.0
-
0.5 Psi=25 m t o r r
t D"=1600cruZ/see
/
0
=
10
i
20
r
aZ
~ - 0 . 1
i
i
30z Icml 40 =
,
,
50
P.~H1.{mI~)
--30
_o
~ 2 o
o
05
Pe =0.3
D :8000cm21sec.
Fig. 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).
2291
C o m p a r i s o n of Fig. 5 and 7 reveals that 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 and s i m u l a tions. M e a s u r e d and c o m p u t e d g r o w t h r a t e profiles
corresponding to c o m p a r a b l e degrees of depletion at
z ~_ 50 a r e found to have 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 results show t h a t high flow rates favor u n i f o r m
g r o w t h rate 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 from the d e creasing a p p a r e n t o r d e r of the reaction 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 expressed b y
the simultaneous occurrence of b o t h the gas flow veloci t y and the i n p u t concentration in the d e n o m i n a t o r of
the factor p.
It should be noted t h a t the calculated c u r v e s for
v e r y low i n p u t values d e v i a t e f r o m the m e a s u r e d
curves. This is due to t h e use of Eq. [1], which is less
valid 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 longer justified. In practice it
was found that 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 small input values, while the e m p l o y e d
m a t h e m a t i c a l relation, Eq. [1], then counts with a half
order. C h a r l i e r (3), who solved the p r o b l e m for firsto r d e r kinetics, showed that in that case the shape of
the g r o w t h rate profiles is no l o n g e r convex (as in Fig.
5), b u t concave, w i t h the lazger p a r t of the d e p l e t i o n
t a k i n g place in the first half of the reactor. This e x plains w h y the last p a r t of the e x p e r i m e n t a l curves
for small 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 decreased so far that the r e a c t i o n a p p r o x i m a t e s first
order, shows a strong d i s c r e p a n c y w i t h the calculated
value, because the depletion 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) have a s l i g h t l y different shape c o m p a r e d to the calculated ones. F o r z > 35
cm the actual 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 the c o m p u t e d profiles. This effect is
a t t r i b u t e d to the a b r u p t l y changing 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 long a n d a short r o w of
wafers were coated, (boat 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 being k e p t constant. The g r o w t h rate at position z = 25 cm, was found
to be 10-15% higher for the large batch. F o r the calculations an infinitely long t u b e and r o w of wafers is
assumed. Consequently this effect is not t a k e n into
account in the model.
Summarizing, one m a y state t h a t using the model
as d e s c r i b e d in this paper, it has b e e n possible to e s t a b lish c l e a r l y the influence of common LPCVD 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 pressure, t o t a l pressure,
a n d the kinetics of the deposition process on the
g r o w t h r a t e profile. Therefore, optimization of LPCVD
processes that are a p p l i e d in IC technology has n o w
become easier.
M a n u s c r i p t s u b m i t t e d Nov. 10, 1981; revised m a n u script 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 will a p p e a r in a Discussion Section to b e p u b l i s h e d in the J u n e 1983 JOURNAL.
A l l discussions for the J u n e 1983 Discussion Section
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.
REFERENCES
1. C. J. H. v a n den B r e k e l and L. J. M. Bollen, J. Cryst.
Growth, 54, 310 (1981).
2. W. A. P. Claassen, J. Bloem, W. G. J. N. V a l k e n b u r g ,
and C. J. H. v a n den Brekel, ibid., I n press.
3. J.-P. Charlier, IEEE Trans. Electron Devices, ed-28,
501 (1981).
4. R. E. O'Malley 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 York (1974).