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An analytical study of the Chemical Vapor Deposition (CVD) processes in a rotating pedestal reactor
Journal of Crystal Growth 77 (1986) 199—208 North-Holland, Amsterdam
199
AN ANALYTICAL STUDY OF THE CHEMICAL VAPOR DEPOSITION (CVD) PROCESSES IN A ROTATING PEDESTAL REACTOR K. CHEN and A.R. MORTAZAVI Department of Mechanical and Indurtrial Engineering University of Utah, Salt Lake City, Utah 84112, USA Received 29 March 1985; manuscript received in final form 24 January 1986
The epitaxial growth and fluid dynamics characteristics of pedestal Chemical Vapor Deposition (CVD) reactors were studied analytically, with emphasis on the effects of susceptor rotation and thermal diffusion on the epitaxial growth rate. The coupled transport equations were solved by numerical integration and a multiple shooting technique. The pressure, velocity, temperature and concentration distributions were calculated and studied for various rotation speeds and operating conditions. The ranges of the process parameters in actual CVD reactors were estimated. It is found that different combinations of the external forced flow and the susceptor rotation have similar temperature and concentration distributions for the Schmidt number around unity when the results are presented in terms of a properly normalized vertical coordinate. The changes in susceptor rotation and external forced flow have nearly identical effects on the epitaxial growth rate for the Schmidt numbers greater than unity. At small Schmidt numbers, the epitaxial growth rate can be enhanced by increasing the forced flow velocity.
1. Introduction Due to the increasing demand for large size and high quality semiconductor wafers, considerable interest has been stimulated in recent years in the problems of gas flow dynamics and heat/mass transfer in Chemical Vapor Deposition (CVD) reactors. While the transport phenomena in horizontal and barrel reactors have been investigated in some detail in the past two decades [1—7],those in pedestal (rotating-disk) CVD reactor (fig. 1) have received little attention [18—20].Although the pedestal reactor has comparatively low wafer capacity, the uniformity of the epitaxial layers grown in this type CVD reactor is the best among the various CVD reactors from the fluid dynamics viewpoint. This is because the three boundary layers (momentum, temperature and concentration) are uniform along the susceptor surface in pedestal reactor as long as the gas flow remains laminar. Tilting of the susceptor to compensate for the reaction depletion is not necessary if the boundary layers are uniform. Therefore, the pedestal type CVD reactors are still widely used in laboratories for fundamental evaluations. The ob-
jective of the present investigation is to study the epitaxial growth rate in pedestal CVD reactors with and without susceptor rotation and the effects of rotation and thermal diffusion on the velocity, temperature and concentration distributions of the gas flows. In most of the pedestal CVD reactors, the mixture of the carrier gas and the reactive species enters the reactor chamber from the center and impinges on the substrates placed horizontally on the susceptor. The susceptor is usually rotated to compensate the non-uniformity of the flow inlet and heating conditions. If the axial flow velocity is high and the diameter of the reactor chamber is considerably larger than the susceptor diameter, the presence of the reactor wall has little influence on the gas flow in a pedestal reactor except in the region very close to the reactor wall. The gas flow in this case can be simplified to a uniform flow toward a rotating disk, as shown in fig. 2. Recent numerical calculations by WahI [19] showed that the velocity and temperature distributions above the susceptor in a pedestal CVD reactor without rotation are very similar to the stagnation flow solutions above an infinite plate, indicating that
0022-0248/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
200
K Chen, A.R. Mortazavi
/ CVD
processes in rotating pedestal reactor
REACTANT GAS INLET z
/i
,,.
WAFER
I
•— V
SUSCEPTOR RF COIL
u~—az
titv
EXHAUST
4
v~~~RXt2
Fig. 2. Selected coordinate system and the corresponding velocity components.
Fig. 1. Pedestal (rotating-disk) CVD reactor.
the assumption of little wall effects in pedestal reactors is valid over a large portion of the susceptor surface. The two limiting cases of this fluid mechanics problem, namely a stagnation flow above a stationary disk and a rotating disk in a quiescent fluid, have been extensively studied in the past. Since the pioneering works of von Karman [21] and Frossling [22], studies of momentum and heat transfer in these two types of flows have been carried out by several researchers both theoretically and experimentally [23—27].The minor errors in von Karman’s calculations were corrected in Cochran’s paper [28]. The problem of an external forced flow against a rotating disk has been considered by Hannah [29]. Schlichting and Truckenbrodt [30]. Millsaps and Poblhausen [31]. and Tifford and Chu [32]. Tifford also investigated the heat transfer by laminar flow from a rotating plate [33]. Unfortunately, no detailed derivations and results were given in Tifford’s papers. Due to the large temperature difference encountered in typical CVD processes, the buoyancy and Soret (thermal diffusion) effects may be very important for the gas flow in a vertical CVD reactor. In the present study, the buoyancy and Soret effects were included in the governing equalion. The coupled continuity, momentum and en-
ergy equations were solved simultaneously by numerical integration and a multiple shooting technique. The rotation and Soret effects on the epitaxial growth rate in a pedestal CVD reactor were examined for different Schmidt numbers.
2. Fonnulation The selected coordinate system and the corresponding velocity components in the present analysis are shown in fig. 2. The fluid is assumed to have constant physical properties but obey the Boussinesq approximation, a necessary assumption for analytical solutions. In fact, the assumption of constant properties is an acceptable approximation for typical CVD processes. Enough experimental and analytical heat convection studies [34] have shown that the constant property results could be applied to gas flows with more than 1000 K temperature variation if an appropriate reference temperature were used. The Navier—Stokes equations for an incompressible axisymmetric flow are: -~---
+
r ~
~
ar
—
+
+
~
0,
-~----
r
Z .~
az
—
r
(1)
K Chen, A.K Mortazavi / CVD processes in rotatingpedestal reactor
1 Bp p Br By
By
(8- 2 Br 1
Br
Bz
r
+
B~u r1 B~ Br + az2
—
/ 82v
1 By
2 8w Br
8w ~
)
‘
B~v Bz
‘~
a,lBp
7u
+
(2) v r
the equation governing concentration distribution becomes
ac
22J~ /
‘B~C 1 ac BT BZ Br2 r Br u—+w—=D(---—+---—-+
g$(T—T~)
IB2w +i’I Br2
+
18w
B2w
r Br
Bz2
——
)
ac
D~B2T 1 BT
~
(3) +
201
B2c
‘~
B2T~
(8)
The momentum_boundary layer thickness is of (4)
the order nate z is normalized ~ft’/(a + I?). as Hence, the vertical coordi-
where u, v and w are the velocity components in the radial, angular and axial directions, respectively. Other variables and parameters are explained in the Nomenclature at the end of this paper. Although the gas flow in actual reactors may be unsteady and turbulent, a laminar flow analysis should provide qualitative guidance in the design of CVD reactors. The energy equation for a two-dimensional laminar flow is
z~= z’/(a + U)/v.
BT BT I 82T 1 BT u—+w——=aj—+———+ Br Bz \ 8r~ r Br
p= —~pa2r2—pv(12+a/2) P(z*), T*(z*) = (T— Tm)/(Ts Tm),
(lOd) (lOe)
C*(z*)
(lof)
B21~) 8z2
(5)
The Dufour effect [34] was neglected in the energy equation. This is because the gas mixtures in typical CVD processes are very dilute. The enhancement in heat transfer due to concentration diffusion is negligible. On the other hand, mass transfer rate enhanced by temperature gradient may be significant because of the large temperature variation involved. The Soret effect should be included in the mass transport equation. The formula for mass transport with the thermal diffusion (Soret) effect [35] included is
1=
=
Inspection of Karman’s results for rotating disk and Frossling’s work for stagnation flow suggests the following nondimensionalization for the dependent variables: = (a + 12) F( z*), (lOa) ~
,
V
w
=
r12 G ( z*), ~v(a + 12) H(z*),
(lob) (lOc)
—
(C
—
Cm),/(Cs
—
Cm).
In terms of these dimensionless variables, the governing equations can be simplified to the following set of ordinary coupled differential equations: H’ + 2F 0, (11)
R 2G2 + F’H F” 2FG +1HG’ G” =0, F2
—
—
R~= 0,
—
(6)
—
NI) cC
—
NXAXB
KTD VT.
All the properties in the above equations are assumed constant and evaluated at the average temperature T0. Based on the above approximations
(12) (13)
(R 1 + R2/2)p’ HH’ + Or T* + H” =0, T*~~_~PrHT*~=0,
(14)
c*~~Sc HC*~ Sr T*~~ =0,
(16)
—
—ND(VC+XAXBKT V ln T).
Applying Taylor’s expansion to In T and dropping the higher order terms the molar flux equation can be approximated as
j
(9)
—
—
(is)
where primes denote differentiation with respect to z~. The parameters involved in these equations are defined below: Grashof number Or =
g$ ( 7
—
Tm) [ v/( a + 12)] 3/2 2
p
(17a)
202
K Chen, A.R. Mortazavi
Prandtl number Pr = v/a, Soret parameter Sr
X~XB
/
CVD processes in rotating pedestal reactor
(17b)
3. Results and discussion
(1 7c)
In the present investigation, the Prandtl number was set to be 0.7, since the Prandtl numbers for most gases, pure or mixed, over a large ternperature range are very close to 0.7. The Grashof
(17d)
R2
=
a/(a
+
12).
(17e)
The parameters R1 and R 2’ respectively, represent the influences of the forced flow and the disk rotation on the fluid flow. When R~= 1 and R2 0, the forced flow vanishes and Karman’s solutions for from a rotating disk in aequations. quiescent Iffluid obtained the resultant R are 1=0 and R2 = l~the disk is not rotated and the governing equations reduce to that for a stagnation flow over a stationary disk. The dimensionless boundary conditions become F(0) = H(0) = P(o) = 0, G(0) = T*(0) = C*(0) 1, G(oo) = T*(oo) = C*(o,o) =0,
(18a) (18b) (18c)
F( no)
(1 8d)
=
R2
-
The governing equations derived above can be transformed into a set of 10 first-order ordinary differential equations. A 4th order Runge—Kutta method was employed to integrate the resultant differential equations numerically. Since the mathematical problem involved is a boundaryvalue problem with more than one velocity boundary conditions to be satisfied at z * no, a multiple shooting technique is required to match the outer boundary conditions at z* no. For simplicity, the outer boundary conditions for velocity and temperature calculations were set at = 10 instead of infinite. Since the momentum boundary layer thickness is of the order a + 12), the calculation depth of the numerical integration is much larger than the boundary layer thickness. Thus, the results should be very close to those in a semi-infinite domain. The calculation depth for the concentration distribution varied from 10 to 100, depending on the value of Sc. —p
—‘
number varied from 0 to 8. Note that the characteristic length of the Grashof number defined in the present study is the momentum boundary layer thickness. If an explit length scale (e.g. disk diameter) were chosen for nondimensionalization, the Or value would be much greater and the dimensionless parameter associated with valthe 2. The selected buoyancy term became Or/Re ues for the Soret parameter in the calculations were 1,0,1 so that both positive and negative effects of thermal diffusion on the epitaxial growth —
-
rate were examined. The parameter values in actua7l rotation disk reactors are estimated and discussed in the appendix. Three different combinations of R1 and R2 were investigated: R1 = 0, R2 = 1; R1 = 0.5, R2 = 05; R~= 1, R2 = 0. The results for any other combinations of the external forced flow and the flow induced by susceptor rotation can be easily interpolated from these three cases. The velocity, temperature, pressure and concentration distributions for different combinations of R1 and R2 are presented in figs. 3a—3c. The angular velocity v = rQG is maximum at the disk surface, and drops to nearly zero at the edge of the momentum boundary layer (z * 5). The magnitude of decreases as r and 12 decrease. The radial velocity u is zero at the disk surface for the requirement of non-slip boundary conditions. If the flow is induced by the rotation of the disk alone (R2 = 0), the radial velocity will reach its maximum value at a point very close to the rotating surface due to the centrifugal force, then drop to zero at the edge of the momentum boundary layer. This velocity profile resembles the velocity profile of natural convection on a vertical plate in a quiescent fluid. If the flow is not solely induced by the disk rotation (R2 * 0), the flow in the radial direction is driven by the centrifugal force and the viscous drag of the external force flow. The influence of the disk rotation is confined within the boundary layer. Thus, the radial velocy
K. Chen, A.R. Mortazavi
/ CVD processes in rotating pedestal reactor
1.00-
0.900.80
203
a
________—P
-
0.70 0.60
-
0.50 0.40 0.30
G
\c*
0.20
0.0
~
3.0
4.0
1.0
5.0
6.0
7.0
8.0
9.0
10.0
z
b
~___...—.--————--—-.——---—--.-F
0.0
2.0
/
1
1.00-
1.0
T*
2.0
3.0
4.0
I
5.0 z
6.0
7.0
I
8.0
9.0
*
~
100
C
o’~~
~
10.0
I
0.0
1.0
2.0
3.0
4.0
5.0 z
5.0
7.0
8.0
9.0
10.0
*
Fig. 3. Velocity, temperature, pressure and concentration distributions for various combinations of the rotation speed and the forced flow velocitywhen Pr~0.7,Gr=Sr=0 and Sc-’l: (a) R 1~=1,R2=0: (b) R1=0.5, R2=0.5: (c) R1=0, R2=1.
ity distribution beyond the boundary layer thickness approaches the potential flow solution of a stagnation flow, The axial velocity w and its slope are zero at the disk surface. Beyond the boundary layer the axial velocity varies linearly with z, as predicted by the potential flow solution of axisymmetric stagnation flow. When R1 = 1, R2 = 0, the axial velocity approaches a constant at z” > 5. The shapes of the dimensionless temperature and concentration distributions are similar to the dimensionless velocity profile in the circumferential direction. The ratios of the momentum boundary layer thickness to the thermal and concentration boundary layer thickness depend on the
Prandtl number and the Schmidt number, respeclively. As can be seen from fig. 3, different combinations of rotation speed and forced flow velocity have similar temperature and concentration distributions when the results are presented in terms of the stretched normal coordinate z *, This observa-ET1 w465 193 m5 lion confirms the selection of ~f a + 12) as the appropriate length scale for z. It can be seen from the z-momentum equation that the inclusion of the buoyancy term affects only the vertical pressure gradient if the velocity boundary conditions are fixed and the flow remains stable. The existence of a buoyance force is equivalent to adding a positive pressure gradient in the vertical direction. For a given forced flow,
204
K Chen, AR. Mortazavi
/
CVI’processes in rotating pedestal reactor
the increase in axial pressure difference due to the deceleration of the approaching flow decreases when the buoyancy effect is taken into consideration. The pressure losses due to the buoyancy effect are shown in fig. 4 for different combinations of R1 and .R2. Inspection of fig. 4 shows that, when the flow is induced by susceptor rotation alone, the axial pressure approaches a constant beyond the boundary layer and the presence of a temperature difference has significant influence on the pressure distribution. As the forced flow increases, the pressure variation along the rotating axis becomes larger and larger, and the buoyancy effect becomes less important to the pressure distribution, The flow field and pressure distributions discussed above are valid if the flow is laminar and free of thermal instability (convection rolls). The critical Or for two-dimensional stagnation flow is around 35 [36]. The present analysis may not be very accurate for too high a Grashof number due to the development of convection rolls above the susceptor. The epitaxial growth rate on the substrate surface is determined by both the concentration diffusion and the thermal diffusion when the Soret effect is taken into account. The dimensionless epitaxial growth rate can be calculated from the temperature and concentration distributions through the following relation: Jv’v/(a + Si) ND(C~ Cm)
~
—
at
z~= 0
=
—
+
~
82*
Sr-~— 32*
~19~ ‘
/
Since C * and T * were assumed to be independent of 6 and r, the epitaxial growth rate is uniform along the susceptor. The dimensionless epitaxial growth rates versus the Schmidt number are plotted in fig. 5 for three different combinations of rotation speed and forced flow velocity. The Soret effect on the epitaxial growth rate is also presented in this figure. A negative Soret parameter Sr enhances the epitaxial growth rate while a positive Sr results in a lower epitaxial growth rate on the substrate surface. The concentration distribution is only slightly affected by thermal diffusion. The in-
20.
/ ~.
/
/
-
~
c.,~
~‘
/
a.
Or
0
*
5
a. o.
R1
*
1.0, R2
R1
=
0.5
*,
R1
=
0.0. R2
•
=
R2
0.0 0.5
-
1.0
8 20 0.0
~.ø
~.o ~.o ~.o ~.o ~.o ~.o ~.o
~.o io.o
Fig. 4. Dependence of axial pressure distributions on buoyancy effect for different combinations of R~and R2.
fluence of the Soret effect on the total epitaxial growth rate decreases as Sc increases. This is because the concentration diffusion rate increases with Sc. Thus, the mass diffusion due to concentration difference dominates at high Schmidt numbers and the thermal diffusion is less important in comparison with the concentration diffusion. Now consider the influences of the susceptor rotation and external forced flow on the epitaxial growth rate. The calculated surface molar flux in fig. 5 shows that the epitaxial growth rates for different combinations of R1 and R2 increase with Sc at different rates. For a given Sc, the epitaxial growth rate of the curve R1 = 0, R2 = 1, is always higher than that of R1 = 1, R2 = 0. An examination of the epitaxial growth rates at different Schmidt numbers reveals that the epitaxial growth rate in a rotating CVD reactor can be enhanced more efficiently by increasing the forced flow velocity, especially at small Schmidt number. In fig. 5 the dimensionless epitaxial growth rate for .R1 =0, R2 = 1 is nearly three times higher than that for R1 = 1, R2 = 0 at Sc 0.1, Sr = 0, but is only 60% higher when Sc is increased to 10. An explanation to this drastic change in the contribution of forced flow to the epitaxial growth can be found in the calculated velocity profiles (figs. 3a—3c). Recall that the ratio of the momenturn boundary layer thickness to the concentration boundary layer thickness is proportional to Sc. At
K Chen, A.R. Mortazavi
/
CVD processes in rotatingpedestal reactor 2.50 -
large Sc, the variation in concentration distribution is confined to a very thin layer in which the velocity profiles for different combinations of R1 and R2 are very similar when presented in terms of z*. The changes in susceptor rotation and the external force flow have nearly identical effects on the concentration distribution in this case. For Sc much less than unity, the concentration boundary layer is much thicker than the momentum boundary layer and the mass transfer rate is determined mainly by the potential flow solution. It is shown in figs. 3a—3c that, beyond the momenturn boundary layer (z* > 5) the velocity of the approaching flow H) is constant for R2 = 0, but increases linearly with z * for R2 > 0. Thus, the existence of an external forced flow can effectively reduce the growth of the concentration boundary layer, resulting in a much higher epitaxial growth rate. Since the temperature and concentration distributions for different combinations of R1 and R2 have nearly identical shapes for Sc and Pr close to or greater than unity, the slopes of the dimensionless temperature and concentration distributions at the substrate surface can be estimated as
2.00
-
1.50
-
205
__________________________________________________________ ~:
Sr=-1.
~
= ~
0. ‘~
6)
*:
Sr-i, @
R~- 0.0
-
0.5. 67
0.5
=
0 0
1 0
R~
=
0,00
—1.00
I —0.52
0.00
I
0.50
1,00
(—
BC * I =
c1
(20a)
Scc2
BT *
Fig. 5. Dimensionless epitaxial growth rate versus Sc for various combinations of R1 and R2 when Pr = 0.7.
Acknowledgement This work was partially supported by the National Science Foundation under grant No. CBT8509311.
Appendix: The ranges of parameter values in actual rotating disk reactors Since the gas mixtures in CVD processes are
~ L*
=C3c,Prc4 (20b) where c1.I~ and range from 0.37 to 0.8 in the present study, the exponents c2 and c4, from previous heat transfer results [37], should be between 1/3 and 1/2. The epitaxial growth rate in typical CVD processes (Pr 0.7, Sc> 1) thus can be calculated from the following equation —
ND(C5
monly encountered in CYD processes is 0.6
The empirical constants c1, c2, c, and c4 are better determined from experiment. But if only modest accuracy is required, the following formula is recommended for epitaxy calculation for Sc around unity. 4 Sc°~4). (22) i —0.6 ND 1v/(a (C, + Si) Coo) (Sr Pr°’
to 1000 rpm [20]. The diameter of the disks used in these experiments is less than 57 mm. The inlet gas velocity, calculated from the reactor dimensions and the gas flow rate given in these previous works [20,40],susceptor, is between 1.5 and 18.5 cm/s. theFrom rotating Hitchman and Curtisabove [20] temperature profiles estimatedthe themeasured thermal boundary layer thickness 8T
=
—
c1
usually dilute, the physical chemical properties those of very the of carrier the gasgases. mixtures The carrier are and very gases close cornto monly used for the fabrication of semiconductor wafers (Si and GaAs) are H2 and N,[3,11]. The Prandtl numbers of these carriers gases are approximately 0.7 over a large temperature range (300—1300 K) [38]. The range of Sc values corn-
(21)
—
Cm) (c, Sr Prc4
—
Scc2).
~Iv/( a + Si)
—
=
—
1
K Chen, A.R. Mortazavi / CVD processes in rotating pedestal reactor
206
Table 1 Typical parameter values for rotating disk CVD reactors Chemical reactions Siepitaxy
Carrier gas 2)
SiH
4—’Si+2H2
i (K) 1300
T,, (K) 300
n
Pr
Sc
Gr
Sr
0.668
0.7
0.6—6
7.8—128
SiH
13 —218
4 inH2: Sr = 0.550 SiCI4 in H2: Sr = 0.608 GaClinH 2: Sr = 0.516 As4 in H2: Sr = 0.529
H2 (MA=
SiCI4 +2 H2 —~ 4 HCI+Si 2) GaAsepitaxy
GaCl+~As4+~H2—’GaAs+HCI
(in fact, the thickness they measure is the conduction thickness [41]) as 7—18 mm for various gas flow rates and rotation speeds. The following values are therefore typical for the momentum boundary layer thickness 6 in a rotating disk reactor: ~
1000
6~6T
0.7
0.6—6
As mentioned in ref. [42], the equation for K,- was developed for isotopic gas mixtures but is sufficiently accurate to yield a first approximation for KT for other gas mixtures. The typical values for Sr, n and other dimensionless parameters in various CVD processes are tabulated in table 1.
Pr”2 [39] =5.9—15mm.
Nomenclature
layer thickness, the typical Gr values for rotating disk reactors are estimated and tabulated in table
a c,—c
1. The properties in the calculations were evaluated at the average temperature ~‘0 = (7 + Tm)/2. It should be kept in mind that Gr values shown in table 1 are based on the momentum boundary layer thickness. These values be diammuch greater if an explicit length scalewould (e.g. disk eter) were used. The Soret parameter (eq. (17c)) for a diffusioncontrolled (C, = 0), dilute gas. mixture (XA 1, XB Cm 1) can be simplified to
C
Forced flow parameter (s1) 5
C* C~ D F g G Gr
2 KT
H
where KT is the thermal-diffusion constant and can be estimated from the following equation [35, 421:
K,. = 1.5
MA (1
—
)
n in the above equation can be estimated from the viscosity—temperature relationship:
C, T”.
Empirical constants Molecular concentration of reactive species Dimensionless concentration Surface concentration 2 s’1) Coefficient of diffusion (cm Dimensionless radial velocity Gravitational acceleration (cm s_2) Dimensionless angular velocity Grashof number (= g$ (T~ Tm) [v/(a + —
12)13/2
-~
=
0.668
1/2
(a+Si) Based on these values for the momentum boundary
Sr
300
H2 (MA=
J
KT MA M 8 n
v~2)
Dimensionless axial velocity Molar flux (mol cm2 s~) Thermal diffusion constant Molecular weight of carrier gas Molecular weight of reactive specie
p
Exponent lationship in the viscosity—temperature reMolecular density of the gas mixture (mol cm _3) Gas pressure (atm)
P
Dimensionless pressure
N
K Chen, A.R. Mortazavi
Pr r Sc Sr T T* T
0
u vw
XA XB z z”
Prandtl number (= v/a) Radial distance (cm) Schmidt number (= v/D) Soret parameter (= [7 C,)]XA Xn KT/To) Temperature (K) Dimensionless temperature
—
Tm
)/(
Average temperature, T0 = (7 (K) Surface temperature (K) Radial velocity (cm S 1) Angular velocity (cm Axial velocity (cm s’)s’ Mole fraction of carrier gas Mole fraction of reactive specie Axial distance from surface (cm) Dimensionless axial distance
/
Cm
CVD processes in rotatingpedestal reactor
[6] E. Fujii, H. Nakamura, K. Haruna and K. Koga, J. Electrochem. Soc. 119 (1972) 1106. [7) R. Takahashi, Y. Koga and K. Sugawara, J. Electrochem. Soc. 119 (1972) 1406. [8] F.W. Dittman, in: Chemical Reactor Engineering II, Ed. H.M. Hulburt (Am. Chem. Soc., Washington, DC, 1974) p. 463.
—
[9] C.W. Manke and L.F. Donaghey, J. Electrochem. Soc. 124 —
Tm)/2 ‘
~
Greek letter symbols a
o °T
p
SI
Thermal diffusivity (erg cm1 s~ K’) Momentum boundary layer thickness (cm) Thermal boundary layer thickness (cm) Dynamic viscosity (g cm~ s~1) Kinematic viscosity (cm2 s 1) Fluid density (g cm’) Rotation rate of disk (rad s’)
Superscript *
207
Dimensionless quantity
(1977) 561. [10] N. Kobayashi, J. Crystal Growth 42 (1978) 357. [11] S. Berkman, V.S. Ban and N. Goldsmith, in: Heteroepitaxial Semiconductor for Electronic Devices, Eds. G.W. and CC. Wung (Springer, New York, 1978) [121 Cullen B.J. Curtis, Physico-Chem. Hydrodyn. 2 (1981) 357.p. 264. [13] [14] [15]
GH. Westphal, D.W. Shaw and R.A. Hartzell, J. Crystal Growth 56 (1982) 324. 43 (1982) C5-235. L.J. Giling, J. Physique M.E. Coltrin, R.J. Kee and J.A. Miller, J. Electrochem. Soc. 131 (1984) 425. [16] F. Rosenberger, 6th American Conf. on Crystal Growth/6th Intern. Conf. on Vapor Growth and Epitaxy, Atlantic City, NJ, 1984. [17] K. Chen, 6th American Conf. on Crystal Growth/6th Intern. Conf. on Vapor Growth and Epitaxy, Atlantic City, NJ, 1984. [18) K. Sugawara, J. Electrochem. Soc. 119 (1972) 1749. [19] G. Wahi, in: Proc. 9th Intern. Coni. on Chemical Vapor Deposition, Eds. MeD. Robinson, C.J.J. den Brekel, G.W. Cullen, J.M. Blocher, Jr. and P. van Rai-Choudhury (Electrochem. Soc., 1984) p. 60. [20] ML. Hitchman and J. Curtis, J. Crystal Growth 60(1982) [21] T. von Karinan, Z. Angew. Math. Mech. 1 (1921) 233. [22] N. Frossling, National Advisory Committee for Aeronautics Tech. Mem. 1432 (1940). [23] Transfer D.L. Oehlbeck and 22 (1979) 601.F.F. Erian, Intern. J. Heat Mass [24] Cz.O. Popiel and L. Boguslawski, Intern. J. Heat Mass
Subscripts s oo
Evaluated at the susceptor surface Evaluated at the free stream state
References
.
(1] S.E. Bradshaw, Intern. J. Electron. 23 (1967) 381. [2] R.W. Andrews, D.M. Rynne and E.G. Wright, Solid State Technol. 12 (1969) 61. [3] F.C. Eversteyn, P.J.W. Severn, C.H.J. van den Brekel and H.L. Peek, J. Electrochem. Soc. 117 (1970) 925. [4] M. Robinson and P. Wong, in: Chemical Vapor Deposition, Eds. J.M. Blocher, Jr. and J.C. Withers (Electrochem. Soc., New York, 1970) p. 127. [5] B.G. Secrest, W.W. Boyd and D.W. Shaw, J. Crystal Growth 10 (1971), 251.
Transfer 18 (1975) 167. [25] E.M. Sparrow and J.L. Gregg, ASME J. Heat Transfer 82 (1960) 294. [26] A. Igushi, K. Komori and R. Izumi, Intern. J. Heat Mass Transfer 22 (1979) 1319. [27] F. Kreith, J.H. Taylor and J.P. Chong, ASME J. Heat Transfer 81(1959) 95. [28] W.G. Cochran, Proc. Cambridge Phil. Soc. 30 (1934) 365. [29] D.M. Hannah, Brit. Aeronautical Research Council Paper No. 10482 (1947). [30] H. Schlichting and E. Truckenbrodt, J. Aeronaut. Sci. 18 (1951), 639. [31] K. Milisaps and K. Pohihausen, J. Aeronaut. Sci. 18 (1951) 354. [32] AN. Tifford and S.T. Chu, J. Aeronaut. Sci. 19 (1952) 284. [33] A.N. Tifford, J. Aeronaut. Sd. 18 (1951) 567. [34) W.M. Kays and2ndM.E. Convective Heat1980) and Mass Transfer, ed. Crawford, (McGraw-Hill, New York, pp. 274, 12, 15.
208
K Chen, A.R. Mortazavi
/ CVD processes in rotatingpedestal reactor
[35] R.D. Present, Kinetic Theory of Gases (McGraw-Hill, New York, 1958) p. 122. [36] K. Chen, M.M. Chen and C.W. Sohn, J. Fluid Mech. 132 (1983) 49. [37] F.M. White, Viscous Fluid Flow (McGraw-Hill, New York, 1974) p. 273. [38] F.P. Incropera and D.P. DeWitt, Introduction to Heat Transfer (Wiley, New York, 1985) p. 683.
[39] M.L. Hitchman, J. Crystal Growth 48 (1980) 394. [40] G. Costrini and JJ. Coleman, J. Appl. Phys. 57 (1985) 2249. [41] D.B. Spalding, J. Fluid Mech. 4 (1958) 22. [42] J.H. Perry, R.H. Perry, C.H. Chilton and S.D. Kirkpatrick, Eds., Chemical Engineer’s Handbook, 4th ed. (McGrawHill, New York, 1963) p. 17—38.
199
AN ANALYTICAL STUDY OF THE CHEMICAL VAPOR DEPOSITION (CVD) PROCESSES IN A ROTATING PEDESTAL REACTOR K. CHEN and A.R. MORTAZAVI Department of Mechanical and Indurtrial Engineering University of Utah, Salt Lake City, Utah 84112, USA Received 29 March 1985; manuscript received in final form 24 January 1986
The epitaxial growth and fluid dynamics characteristics of pedestal Chemical Vapor Deposition (CVD) reactors were studied analytically, with emphasis on the effects of susceptor rotation and thermal diffusion on the epitaxial growth rate. The coupled transport equations were solved by numerical integration and a multiple shooting technique. The pressure, velocity, temperature and concentration distributions were calculated and studied for various rotation speeds and operating conditions. The ranges of the process parameters in actual CVD reactors were estimated. It is found that different combinations of the external forced flow and the susceptor rotation have similar temperature and concentration distributions for the Schmidt number around unity when the results are presented in terms of a properly normalized vertical coordinate. The changes in susceptor rotation and external forced flow have nearly identical effects on the epitaxial growth rate for the Schmidt numbers greater than unity. At small Schmidt numbers, the epitaxial growth rate can be enhanced by increasing the forced flow velocity.
1. Introduction Due to the increasing demand for large size and high quality semiconductor wafers, considerable interest has been stimulated in recent years in the problems of gas flow dynamics and heat/mass transfer in Chemical Vapor Deposition (CVD) reactors. While the transport phenomena in horizontal and barrel reactors have been investigated in some detail in the past two decades [1—7],those in pedestal (rotating-disk) CVD reactor (fig. 1) have received little attention [18—20].Although the pedestal reactor has comparatively low wafer capacity, the uniformity of the epitaxial layers grown in this type CVD reactor is the best among the various CVD reactors from the fluid dynamics viewpoint. This is because the three boundary layers (momentum, temperature and concentration) are uniform along the susceptor surface in pedestal reactor as long as the gas flow remains laminar. Tilting of the susceptor to compensate for the reaction depletion is not necessary if the boundary layers are uniform. Therefore, the pedestal type CVD reactors are still widely used in laboratories for fundamental evaluations. The ob-
jective of the present investigation is to study the epitaxial growth rate in pedestal CVD reactors with and without susceptor rotation and the effects of rotation and thermal diffusion on the velocity, temperature and concentration distributions of the gas flows. In most of the pedestal CVD reactors, the mixture of the carrier gas and the reactive species enters the reactor chamber from the center and impinges on the substrates placed horizontally on the susceptor. The susceptor is usually rotated to compensate the non-uniformity of the flow inlet and heating conditions. If the axial flow velocity is high and the diameter of the reactor chamber is considerably larger than the susceptor diameter, the presence of the reactor wall has little influence on the gas flow in a pedestal reactor except in the region very close to the reactor wall. The gas flow in this case can be simplified to a uniform flow toward a rotating disk, as shown in fig. 2. Recent numerical calculations by WahI [19] showed that the velocity and temperature distributions above the susceptor in a pedestal CVD reactor without rotation are very similar to the stagnation flow solutions above an infinite plate, indicating that
0022-0248/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
200
K Chen, A.R. Mortazavi
/ CVD
processes in rotating pedestal reactor
REACTANT GAS INLET z
/i
,,.
WAFER
I
•— V
SUSCEPTOR RF COIL
u~—az
titv
EXHAUST
4
v~~~RXt2
Fig. 2. Selected coordinate system and the corresponding velocity components.
Fig. 1. Pedestal (rotating-disk) CVD reactor.
the assumption of little wall effects in pedestal reactors is valid over a large portion of the susceptor surface. The two limiting cases of this fluid mechanics problem, namely a stagnation flow above a stationary disk and a rotating disk in a quiescent fluid, have been extensively studied in the past. Since the pioneering works of von Karman [21] and Frossling [22], studies of momentum and heat transfer in these two types of flows have been carried out by several researchers both theoretically and experimentally [23—27].The minor errors in von Karman’s calculations were corrected in Cochran’s paper [28]. The problem of an external forced flow against a rotating disk has been considered by Hannah [29]. Schlichting and Truckenbrodt [30]. Millsaps and Poblhausen [31]. and Tifford and Chu [32]. Tifford also investigated the heat transfer by laminar flow from a rotating plate [33]. Unfortunately, no detailed derivations and results were given in Tifford’s papers. Due to the large temperature difference encountered in typical CVD processes, the buoyancy and Soret (thermal diffusion) effects may be very important for the gas flow in a vertical CVD reactor. In the present study, the buoyancy and Soret effects were included in the governing equalion. The coupled continuity, momentum and en-
ergy equations were solved simultaneously by numerical integration and a multiple shooting technique. The rotation and Soret effects on the epitaxial growth rate in a pedestal CVD reactor were examined for different Schmidt numbers.
2. Fonnulation The selected coordinate system and the corresponding velocity components in the present analysis are shown in fig. 2. The fluid is assumed to have constant physical properties but obey the Boussinesq approximation, a necessary assumption for analytical solutions. In fact, the assumption of constant properties is an acceptable approximation for typical CVD processes. Enough experimental and analytical heat convection studies [34] have shown that the constant property results could be applied to gas flows with more than 1000 K temperature variation if an appropriate reference temperature were used. The Navier—Stokes equations for an incompressible axisymmetric flow are: -~---
+
r ~
~
ar
—
+
+
~
0,
-~----
r
Z .~
az
—
r
(1)
K Chen, A.K Mortazavi / CVD processes in rotatingpedestal reactor
1 Bp p Br By
By
(8- 2 Br 1
Br
Bz
r
+
B~u r1 B~ Br + az2
—
/ 82v
1 By
2 8w Br
8w ~
)
‘
B~v Bz
‘~
a,lBp
7u
+
(2) v r
the equation governing concentration distribution becomes
ac
22J~ /
‘B~C 1 ac BT BZ Br2 r Br u—+w—=D(---—+---—-+
g$(T—T~)
IB2w +i’I Br2
+
18w
B2w
r Br
Bz2
——
)
ac
D~B2T 1 BT
~
(3) +
201
B2c
‘~
B2T~
(8)
The momentum_boundary layer thickness is of (4)
the order nate z is normalized ~ft’/(a + I?). as Hence, the vertical coordi-
where u, v and w are the velocity components in the radial, angular and axial directions, respectively. Other variables and parameters are explained in the Nomenclature at the end of this paper. Although the gas flow in actual reactors may be unsteady and turbulent, a laminar flow analysis should provide qualitative guidance in the design of CVD reactors. The energy equation for a two-dimensional laminar flow is
z~= z’/(a + U)/v.
BT BT I 82T 1 BT u—+w——=aj—+———+ Br Bz \ 8r~ r Br
p= —~pa2r2—pv(12+a/2) P(z*), T*(z*) = (T— Tm)/(Ts Tm),
(lOd) (lOe)
C*(z*)
(lof)
B21~) 8z2
(5)
The Dufour effect [34] was neglected in the energy equation. This is because the gas mixtures in typical CVD processes are very dilute. The enhancement in heat transfer due to concentration diffusion is negligible. On the other hand, mass transfer rate enhanced by temperature gradient may be significant because of the large temperature variation involved. The Soret effect should be included in the mass transport equation. The formula for mass transport with the thermal diffusion (Soret) effect [35] included is
1=
=
Inspection of Karman’s results for rotating disk and Frossling’s work for stagnation flow suggests the following nondimensionalization for the dependent variables: = (a + 12) F( z*), (lOa) ~
,
V
w
=
r12 G ( z*), ~v(a + 12) H(z*),
(lob) (lOc)
—
(C
—
Cm),/(Cs
—
Cm).
In terms of these dimensionless variables, the governing equations can be simplified to the following set of ordinary coupled differential equations: H’ + 2F 0, (11)
R 2G2 + F’H F” 2FG +1HG’ G” =0, F2
—
—
R~= 0,
—
(6)
—
NI) cC
—
NXAXB
KTD VT.
All the properties in the above equations are assumed constant and evaluated at the average temperature T0. Based on the above approximations
(12) (13)
(R 1 + R2/2)p’ HH’ + Or T* + H” =0, T*~~_~PrHT*~=0,
(14)
c*~~Sc HC*~ Sr T*~~ =0,
(16)
—
—ND(VC+XAXBKT V ln T).
Applying Taylor’s expansion to In T and dropping the higher order terms the molar flux equation can be approximated as
j
(9)
—
—
(is)
where primes denote differentiation with respect to z~. The parameters involved in these equations are defined below: Grashof number Or =
g$ ( 7
—
Tm) [ v/( a + 12)] 3/2 2
p
(17a)
202
K Chen, A.R. Mortazavi
Prandtl number Pr = v/a, Soret parameter Sr
X~XB
/
CVD processes in rotating pedestal reactor
(17b)
3. Results and discussion
(1 7c)
In the present investigation, the Prandtl number was set to be 0.7, since the Prandtl numbers for most gases, pure or mixed, over a large ternperature range are very close to 0.7. The Grashof
(17d)
R2
=
a/(a
+
12).
(17e)
The parameters R1 and R 2’ respectively, represent the influences of the forced flow and the disk rotation on the fluid flow. When R~= 1 and R2 0, the forced flow vanishes and Karman’s solutions for from a rotating disk in aequations. quiescent Iffluid obtained the resultant R are 1=0 and R2 = l~the disk is not rotated and the governing equations reduce to that for a stagnation flow over a stationary disk. The dimensionless boundary conditions become F(0) = H(0) = P(o) = 0, G(0) = T*(0) = C*(0) 1, G(oo) = T*(oo) = C*(o,o) =0,
(18a) (18b) (18c)
F( no)
(1 8d)
=
R2
-
The governing equations derived above can be transformed into a set of 10 first-order ordinary differential equations. A 4th order Runge—Kutta method was employed to integrate the resultant differential equations numerically. Since the mathematical problem involved is a boundaryvalue problem with more than one velocity boundary conditions to be satisfied at z * no, a multiple shooting technique is required to match the outer boundary conditions at z* no. For simplicity, the outer boundary conditions for velocity and temperature calculations were set at = 10 instead of infinite. Since the momentum boundary layer thickness is of the order a + 12), the calculation depth of the numerical integration is much larger than the boundary layer thickness. Thus, the results should be very close to those in a semi-infinite domain. The calculation depth for the concentration distribution varied from 10 to 100, depending on the value of Sc. —p
—‘
number varied from 0 to 8. Note that the characteristic length of the Grashof number defined in the present study is the momentum boundary layer thickness. If an explit length scale (e.g. disk diameter) were chosen for nondimensionalization, the Or value would be much greater and the dimensionless parameter associated with valthe 2. The selected buoyancy term became Or/Re ues for the Soret parameter in the calculations were 1,0,1 so that both positive and negative effects of thermal diffusion on the epitaxial growth —
-
rate were examined. The parameter values in actua7l rotation disk reactors are estimated and discussed in the appendix. Three different combinations of R1 and R2 were investigated: R1 = 0, R2 = 1; R1 = 0.5, R2 = 05; R~= 1, R2 = 0. The results for any other combinations of the external forced flow and the flow induced by susceptor rotation can be easily interpolated from these three cases. The velocity, temperature, pressure and concentration distributions for different combinations of R1 and R2 are presented in figs. 3a—3c. The angular velocity v = rQG is maximum at the disk surface, and drops to nearly zero at the edge of the momentum boundary layer (z * 5). The magnitude of decreases as r and 12 decrease. The radial velocity u is zero at the disk surface for the requirement of non-slip boundary conditions. If the flow is induced by the rotation of the disk alone (R2 = 0), the radial velocity will reach its maximum value at a point very close to the rotating surface due to the centrifugal force, then drop to zero at the edge of the momentum boundary layer. This velocity profile resembles the velocity profile of natural convection on a vertical plate in a quiescent fluid. If the flow is not solely induced by the disk rotation (R2 * 0), the flow in the radial direction is driven by the centrifugal force and the viscous drag of the external force flow. The influence of the disk rotation is confined within the boundary layer. Thus, the radial velocy
K. Chen, A.R. Mortazavi
/ CVD processes in rotating pedestal reactor
1.00-
0.900.80
203
a
________—P
-
0.70 0.60
-
0.50 0.40 0.30
G
\c*
0.20
0.0
~
3.0
4.0
1.0
5.0
6.0
7.0
8.0
9.0
10.0
z
b
~___...—.--————--—-.——---—--.-F
0.0
2.0
/
1
1.00-
1.0
T*
2.0
3.0
4.0
I
5.0 z
6.0
7.0
I
8.0
9.0
*
~
100
C
o’~~
~
10.0
I
0.0
1.0
2.0
3.0
4.0
5.0 z
5.0
7.0
8.0
9.0
10.0
*
Fig. 3. Velocity, temperature, pressure and concentration distributions for various combinations of the rotation speed and the forced flow velocitywhen Pr~0.7,Gr=Sr=0 and Sc-’l: (a) R 1~=1,R2=0: (b) R1=0.5, R2=0.5: (c) R1=0, R2=1.
ity distribution beyond the boundary layer thickness approaches the potential flow solution of a stagnation flow, The axial velocity w and its slope are zero at the disk surface. Beyond the boundary layer the axial velocity varies linearly with z, as predicted by the potential flow solution of axisymmetric stagnation flow. When R1 = 1, R2 = 0, the axial velocity approaches a constant at z” > 5. The shapes of the dimensionless temperature and concentration distributions are similar to the dimensionless velocity profile in the circumferential direction. The ratios of the momentum boundary layer thickness to the thermal and concentration boundary layer thickness depend on the
Prandtl number and the Schmidt number, respeclively. As can be seen from fig. 3, different combinations of rotation speed and forced flow velocity have similar temperature and concentration distributions when the results are presented in terms of the stretched normal coordinate z *, This observa-ET1 w465 193 m5 lion confirms the selection of ~f a + 12) as the appropriate length scale for z. It can be seen from the z-momentum equation that the inclusion of the buoyancy term affects only the vertical pressure gradient if the velocity boundary conditions are fixed and the flow remains stable. The existence of a buoyance force is equivalent to adding a positive pressure gradient in the vertical direction. For a given forced flow,
204
K Chen, AR. Mortazavi
/
CVI’processes in rotating pedestal reactor
the increase in axial pressure difference due to the deceleration of the approaching flow decreases when the buoyancy effect is taken into consideration. The pressure losses due to the buoyancy effect are shown in fig. 4 for different combinations of R1 and .R2. Inspection of fig. 4 shows that, when the flow is induced by susceptor rotation alone, the axial pressure approaches a constant beyond the boundary layer and the presence of a temperature difference has significant influence on the pressure distribution. As the forced flow increases, the pressure variation along the rotating axis becomes larger and larger, and the buoyancy effect becomes less important to the pressure distribution, The flow field and pressure distributions discussed above are valid if the flow is laminar and free of thermal instability (convection rolls). The critical Or for two-dimensional stagnation flow is around 35 [36]. The present analysis may not be very accurate for too high a Grashof number due to the development of convection rolls above the susceptor. The epitaxial growth rate on the substrate surface is determined by both the concentration diffusion and the thermal diffusion when the Soret effect is taken into account. The dimensionless epitaxial growth rate can be calculated from the temperature and concentration distributions through the following relation: Jv’v/(a + Si) ND(C~ Cm)
~
—
at
z~= 0
=
—
+
~
82*
Sr-~— 32*
~19~ ‘
/
Since C * and T * were assumed to be independent of 6 and r, the epitaxial growth rate is uniform along the susceptor. The dimensionless epitaxial growth rates versus the Schmidt number are plotted in fig. 5 for three different combinations of rotation speed and forced flow velocity. The Soret effect on the epitaxial growth rate is also presented in this figure. A negative Soret parameter Sr enhances the epitaxial growth rate while a positive Sr results in a lower epitaxial growth rate on the substrate surface. The concentration distribution is only slightly affected by thermal diffusion. The in-
20.
/ ~.
/
/
-
~
c.,~
~‘
/
a.
Or
0
*
5
a. o.
R1
*
1.0, R2
R1
=
0.5
*,
R1
=
0.0. R2
•
=
R2
0.0 0.5
-
1.0
8 20 0.0
~.ø
~.o ~.o ~.o ~.o ~.o ~.o ~.o
~.o io.o
Fig. 4. Dependence of axial pressure distributions on buoyancy effect for different combinations of R~and R2.
fluence of the Soret effect on the total epitaxial growth rate decreases as Sc increases. This is because the concentration diffusion rate increases with Sc. Thus, the mass diffusion due to concentration difference dominates at high Schmidt numbers and the thermal diffusion is less important in comparison with the concentration diffusion. Now consider the influences of the susceptor rotation and external forced flow on the epitaxial growth rate. The calculated surface molar flux in fig. 5 shows that the epitaxial growth rates for different combinations of R1 and R2 increase with Sc at different rates. For a given Sc, the epitaxial growth rate of the curve R1 = 0, R2 = 1, is always higher than that of R1 = 1, R2 = 0. An examination of the epitaxial growth rates at different Schmidt numbers reveals that the epitaxial growth rate in a rotating CVD reactor can be enhanced more efficiently by increasing the forced flow velocity, especially at small Schmidt number. In fig. 5 the dimensionless epitaxial growth rate for .R1 =0, R2 = 1 is nearly three times higher than that for R1 = 1, R2 = 0 at Sc 0.1, Sr = 0, but is only 60% higher when Sc is increased to 10. An explanation to this drastic change in the contribution of forced flow to the epitaxial growth can be found in the calculated velocity profiles (figs. 3a—3c). Recall that the ratio of the momenturn boundary layer thickness to the concentration boundary layer thickness is proportional to Sc. At
K Chen, A.R. Mortazavi
/
CVD processes in rotatingpedestal reactor 2.50 -
large Sc, the variation in concentration distribution is confined to a very thin layer in which the velocity profiles for different combinations of R1 and R2 are very similar when presented in terms of z*. The changes in susceptor rotation and the external force flow have nearly identical effects on the concentration distribution in this case. For Sc much less than unity, the concentration boundary layer is much thicker than the momentum boundary layer and the mass transfer rate is determined mainly by the potential flow solution. It is shown in figs. 3a—3c that, beyond the momenturn boundary layer (z* > 5) the velocity of the approaching flow H) is constant for R2 = 0, but increases linearly with z * for R2 > 0. Thus, the existence of an external forced flow can effectively reduce the growth of the concentration boundary layer, resulting in a much higher epitaxial growth rate. Since the temperature and concentration distributions for different combinations of R1 and R2 have nearly identical shapes for Sc and Pr close to or greater than unity, the slopes of the dimensionless temperature and concentration distributions at the substrate surface can be estimated as
2.00
-
1.50
-
205
__________________________________________________________ ~:
Sr=-1.
~
= ~
0. ‘~
6)
*:
Sr-i, @
R~- 0.0
-
0.5. 67
0.5
=
0 0
1 0
R~
=
0,00
—1.00
I —0.52
0.00
I
0.50
1,00
(—
BC * I =
c1
(20a)
Scc2
BT *
Fig. 5. Dimensionless epitaxial growth rate versus Sc for various combinations of R1 and R2 when Pr = 0.7.
Acknowledgement This work was partially supported by the National Science Foundation under grant No. CBT8509311.
Appendix: The ranges of parameter values in actual rotating disk reactors Since the gas mixtures in CVD processes are
~ L*
=C3c,Prc4 (20b) where c1.I~ and range from 0.37 to 0.8 in the present study, the exponents c2 and c4, from previous heat transfer results [37], should be between 1/3 and 1/2. The epitaxial growth rate in typical CVD processes (Pr 0.7, Sc> 1) thus can be calculated from the following equation —
ND(C5
monly encountered in CYD processes is 0.6
The empirical constants c1, c2, c, and c4 are better determined from experiment. But if only modest accuracy is required, the following formula is recommended for epitaxy calculation for Sc around unity. 4 Sc°~4). (22) i —0.6 ND 1v/(a (C, + Si) Coo) (Sr Pr°’
to 1000 rpm [20]. The diameter of the disks used in these experiments is less than 57 mm. The inlet gas velocity, calculated from the reactor dimensions and the gas flow rate given in these previous works [20,40],susceptor, is between 1.5 and 18.5 cm/s. theFrom rotating Hitchman and Curtisabove [20] temperature profiles estimatedthe themeasured thermal boundary layer thickness 8T
=
—
c1
usually dilute, the physical chemical properties those of very the of carrier the gasgases. mixtures The carrier are and very gases close cornto monly used for the fabrication of semiconductor wafers (Si and GaAs) are H2 and N,[3,11]. The Prandtl numbers of these carriers gases are approximately 0.7 over a large temperature range (300—1300 K) [38]. The range of Sc values corn-
(21)
—
Cm) (c, Sr Prc4
—
Scc2).
~Iv/( a + Si)
—
=
—
1
K Chen, A.R. Mortazavi / CVD processes in rotating pedestal reactor
206
Table 1 Typical parameter values for rotating disk CVD reactors Chemical reactions Siepitaxy
Carrier gas 2)
SiH
4—’Si+2H2
i (K) 1300
T,, (K) 300
n
Pr
Sc
Gr
Sr
0.668
0.7
0.6—6
7.8—128
SiH
13 —218
4 inH2: Sr = 0.550 SiCI4 in H2: Sr = 0.608 GaClinH 2: Sr = 0.516 As4 in H2: Sr = 0.529
H2 (MA=
SiCI4 +2 H2 —~ 4 HCI+Si 2) GaAsepitaxy
GaCl+~As4+~H2—’GaAs+HCI
(in fact, the thickness they measure is the conduction thickness [41]) as 7—18 mm for various gas flow rates and rotation speeds. The following values are therefore typical for the momentum boundary layer thickness 6 in a rotating disk reactor: ~
1000
6~6T
0.7
0.6—6
As mentioned in ref. [42], the equation for K,- was developed for isotopic gas mixtures but is sufficiently accurate to yield a first approximation for KT for other gas mixtures. The typical values for Sr, n and other dimensionless parameters in various CVD processes are tabulated in table 1.
Pr”2 [39] =5.9—15mm.
Nomenclature
layer thickness, the typical Gr values for rotating disk reactors are estimated and tabulated in table
a c,—c
1. The properties in the calculations were evaluated at the average temperature ~‘0 = (7 + Tm)/2. It should be kept in mind that Gr values shown in table 1 are based on the momentum boundary layer thickness. These values be diammuch greater if an explicit length scalewould (e.g. disk eter) were used. The Soret parameter (eq. (17c)) for a diffusioncontrolled (C, = 0), dilute gas. mixture (XA 1, XB Cm 1) can be simplified to
C
Forced flow parameter (s1) 5
C* C~ D F g G Gr
2 KT
H
where KT is the thermal-diffusion constant and can be estimated from the following equation [35, 421:
K,. = 1.5
MA (1
—
)
n in the above equation can be estimated from the viscosity—temperature relationship:
C, T”.
Empirical constants Molecular concentration of reactive species Dimensionless concentration Surface concentration 2 s’1) Coefficient of diffusion (cm Dimensionless radial velocity Gravitational acceleration (cm s_2) Dimensionless angular velocity Grashof number (= g$ (T~ Tm) [v/(a + —
12)13/2
-~
=
0.668
1/2
(a+Si) Based on these values for the momentum boundary
Sr
300
H2 (MA=
J
KT MA M 8 n
v~2)
Dimensionless axial velocity Molar flux (mol cm2 s~) Thermal diffusion constant Molecular weight of carrier gas Molecular weight of reactive specie
p
Exponent lationship in the viscosity—temperature reMolecular density of the gas mixture (mol cm _3) Gas pressure (atm)
P
Dimensionless pressure
N
K Chen, A.R. Mortazavi
Pr r Sc Sr T T* T
0
u vw
XA XB z z”
Prandtl number (= v/a) Radial distance (cm) Schmidt number (= v/D) Soret parameter (= [7 C,)]XA Xn KT/To) Temperature (K) Dimensionless temperature
—
Tm
)/(
Average temperature, T0 = (7 (K) Surface temperature (K) Radial velocity (cm S 1) Angular velocity (cm Axial velocity (cm s’)s’ Mole fraction of carrier gas Mole fraction of reactive specie Axial distance from surface (cm) Dimensionless axial distance
/
Cm
CVD processes in rotatingpedestal reactor
[6] E. Fujii, H. Nakamura, K. Haruna and K. Koga, J. Electrochem. Soc. 119 (1972) 1106. [7) R. Takahashi, Y. Koga and K. Sugawara, J. Electrochem. Soc. 119 (1972) 1406. [8] F.W. Dittman, in: Chemical Reactor Engineering II, Ed. H.M. Hulburt (Am. Chem. Soc., Washington, DC, 1974) p. 463.
—
[9] C.W. Manke and L.F. Donaghey, J. Electrochem. Soc. 124 —
Tm)/2 ‘
~
Greek letter symbols a
o °T
p
SI
Thermal diffusivity (erg cm1 s~ K’) Momentum boundary layer thickness (cm) Thermal boundary layer thickness (cm) Dynamic viscosity (g cm~ s~1) Kinematic viscosity (cm2 s 1) Fluid density (g cm’) Rotation rate of disk (rad s’)
Superscript *
207
Dimensionless quantity
(1977) 561. [10] N. Kobayashi, J. Crystal Growth 42 (1978) 357. [11] S. Berkman, V.S. Ban and N. Goldsmith, in: Heteroepitaxial Semiconductor for Electronic Devices, Eds. G.W. and CC. Wung (Springer, New York, 1978) [121 Cullen B.J. Curtis, Physico-Chem. Hydrodyn. 2 (1981) 357.p. 264. [13] [14] [15]
GH. Westphal, D.W. Shaw and R.A. Hartzell, J. Crystal Growth 56 (1982) 324. 43 (1982) C5-235. L.J. Giling, J. Physique M.E. Coltrin, R.J. Kee and J.A. Miller, J. Electrochem. Soc. 131 (1984) 425. [16] F. Rosenberger, 6th American Conf. on Crystal Growth/6th Intern. Conf. on Vapor Growth and Epitaxy, Atlantic City, NJ, 1984. [17] K. Chen, 6th American Conf. on Crystal Growth/6th Intern. Conf. on Vapor Growth and Epitaxy, Atlantic City, NJ, 1984. [18) K. Sugawara, J. Electrochem. Soc. 119 (1972) 1749. [19] G. Wahi, in: Proc. 9th Intern. Coni. on Chemical Vapor Deposition, Eds. MeD. Robinson, C.J.J. den Brekel, G.W. Cullen, J.M. Blocher, Jr. and P. van Rai-Choudhury (Electrochem. Soc., 1984) p. 60. [20] ML. Hitchman and J. Curtis, J. Crystal Growth 60(1982) [21] T. von Karinan, Z. Angew. Math. Mech. 1 (1921) 233. [22] N. Frossling, National Advisory Committee for Aeronautics Tech. Mem. 1432 (1940). [23] Transfer D.L. Oehlbeck and 22 (1979) 601.F.F. Erian, Intern. J. Heat Mass [24] Cz.O. Popiel and L. Boguslawski, Intern. J. Heat Mass
Subscripts s oo
Evaluated at the susceptor surface Evaluated at the free stream state
References
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208
K Chen, A.R. Mortazavi
/ CVD processes in rotatingpedestal reactor
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