Ind. Eng. Chem. Res. 1989, 28, 1162-1170
1162
Cookson, D. J.; Smith, B. E. Determination of Structural Characteristics of Saturates from Diesel and Kerosene Fuels by Carbon-13 Nuclear Magnetic Resonance Spectrometry. Anal. Chem. 1985, 57, 864-871. Cookson, D. J.; Smith, B. E. One- and Two-Dimensional NMR Methods for Elucidating Structural Characteristics of Aromatic Fractions from Petroleum and Synthetic Fuels. Energy Fuels 1987, 1, 111-120. Delpuech, J.-J.;Nicole, D.; Daubenfeld, J.-M.; Boubel, J.-C. Method to Evaluate Benzonaphthenic Carbons and Donatable Hydrogens in Fossil Fuels. Fuel 1985, 64, 325-334. Delpuech, J.-J.;Nicole, D.; LeRoux, M.; Chiche, P.; Pregermaine, S. Coal Liquefaction solvents: An NMR Analysis of a Recycle Oil Over Successive Passes. Fuel 1986, 65, 1600-1607. Gates, B. C.; Katzer, J. R.; Schuit, G. C. A. Chemistry of Catalytic Processes; McGraw-Hill: New York, 1979; p 392. Gillet, S.; Rubini, P.; Delpuech, J.-J.; Escalier, J.-C.; Valentin, P. Quantitative Carbon-13 and Proton Nuclear Magnetic Resonance Spectroscopy of Crude Oil and Petroleum Products. Fuel 1981, 60, 221-225. Haynes, H. W., Jr. Improved Catalysts for Coal Liquefaction. Final Report USDOE PC/70812-13, April 1988. Haynes, H. W., Jr.; Parcher, J. F.; Heimer, N. E. Hydrocracking Polycyclic Hydrocarbons over a Dual-Functional Zeolite (Faujasite)-Based Catalyst. Ind. Eng. Chem. Process Des. Dev. 1983,22, 401-409. Kamiya, Y.; Sata, H.; Yao, T. Effect of Phenolic Compounds on Liquefaction of Coal in the Presence of Hydrogen Donor Solvent. Fuel 1978,57, 681-685. Ladner, W. R.; Snape, C. E. Application of Quantitative 13C NMR Spectroscopy to Coal-Derived Material. Fuel 1978,57, 658-662. Mahoney, J. A.; Schwartz, M. M.; Wittrig, T. S. Advanced Coal Liquefaction Development. Annual Report, USDOE PC/40009T5, Dec 1982. McCormick, R. L.; Baker, J. R.; Haynes, H. W., Jr.; Malhotra, R.
Surface Acidity Studied by the Temperature-Programmed Desorption of tert-Butylamine. Energy Fuels 1988, 2, 740-743. McCormick, R. L.; King, J. A.; King, T. R.; Haynes, H. W., Jr. Influence of Support on the Performance of Coal Liquid Hydrotreating Catalysts. Ind. Eng. Chem. Res. 1989, in press. Mieville, R. L.; Meyers, B. L. Measuring Acidity by TemperatureProgrammed Desorption. J . Catal. 1982, 74, 196-198. Miller, R. L.; Silver, H. F.; Hurtubise, R. J. Upgrading of Recycle Solvent Used in the Direct Liquefaction of Wyodak Coal. Ind. Eng. Chem. Process Des. Dev. 1982,21, 170-173. Nelson, H. C.; Lussier, R. J.; Still, M. E. An Estimate of Surface Acidity in Amorphous Catalysts From Temperature-Programmed Desorption Measurements. A Simple Tool for Catalyst Characterization. Appl. Catal. 1983, 7, 113-121. Netzel, D. A. Quantitation of Carbon Types Using DEPT/QUAT NMR Pulse Sequences: Application to Fossil-Fuel-Derived Oils. Anal. Chem. 1987,59, 1775-1779. Netzel, D. A., Guffy, E. D. NMR and GC/MS Investigation of the Saturate Fractions From the Cerro Negro Heavy Petroleum Crude. Energy Fuels 1989, in press. Pegg, D. J.; Doddrell, D. M.; Bendall, M. R. Proton-Polarization Transfer Enhancement of a Heteronuclear Spin Multiplet with Preservation of Phase Coherency and Relative Component Intensities. J . Chem. Phys. 1982, 77, 2745-2752. Ruberto, R. Fate of Solvents in the Non-Catalytic Depolemerization of Kentucky Bituminous Coal. Fuel Process. Technol. 1984, 3, 7-24. Spencer, D. EPRI J . 1987, 12, 40. Suarez, W.; Dumesic, J. A.; Hill, C. G., Jr. Acidic Properties of Molybdena-Alumina for Different Extents of Reduction: Infrared and Gravimetric Studies of Adsorbed Pyridine. J . Catal. 1985, 94, 408-421. Received f o r review August 25, 1988 Accepted April 7, 1989
Optimization of a Low-Pressure Chemical Vapor Deposition Reactor for the Deposition of Thin Films Tarshish Setahad,+Isaac Trachtenberg, B. Wayne Bequette,t and Thomas F. Edgar* Department of Chemical Engineering, The University of Texas, Austin, Texas 78712
A simulation model for the low-pressure chemical vapor deposition of polycrystalline silicon using a silane-hydrogen mixture in a multiwafer batch record has been developed. This model was employed to study the effects of temperature, flow parameters, reactor geometry, and wafer size upon the process, particularly the uniformity of silicon deposition. Potential improvements in the system performance were determined by utilizing optimum temperature staging and reactant injection schemes. The results also showed that nonuniform wafer spacing can improve deposition uniformity and wafer throughput while decreasing the process sensitivity to reactant flow rate variations. The manufacture of microelectronic devices involves the sequencing of processes involving thin film deposition, patterning, and doping. The formation of the film is performed by a variety of techniques including physical and/or chemical processes (Jensen, 1987). One of the most versatile of these methods is chemical vapor deposition (CVD). This process involves reacting flowing gases on a substrate to form the desired film. Energy for the reaction is provided by heat or by a plasma. CVD requires the diffusion of gaseous reactants to the hot substrate and adsorption, reaction, desorption, and diffusion of gaseous products back into the bulk gas, yielding a film on the substrate. Present address: Signetics, Albuquerque, N M 87184.
* Present address: Department of Chemical Engineering, RPI,
Schenectady, NY 12180-3590.
The configuration commonly used for most depositions has wafers horizontally stacked in a tube. This lowpressure chemical vapor deposition (LPCVD) reactor, shown in Figure 1, allows a large number of wafers to be processed, with good film thickness and composition uniformity (Rosler, 1977). The operation of LPCVD systems is normally conducted in an empirical fashion because of lack of knowledge of the process fundamentals. Since the electronics industry is now beginning to use larger diameter wafers along with tighter specifications, there is an incentive to make. use of fundamental process models to optimize the design and operation of LPCVD systems. The purpose of this paper is to make use of a fundamental LPCVD model to determine the sensitivity of wafer film uniformity to process operating parameters, such as reactor temperature distribution, wafer spacing, and intermediate feed injection rates. Another goal is to compute the optimal design and
0888-5885/89/2628-1162$01.50/0 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1163 the interwafer region between any two wafers was reasonable for a Thiele modulus less than 10. Velander and Pressure White (1987) considered induced convection effects due Sensor to the reaction stoichiometry and developed a dimen? sionless number to predict when these effects could cause intrawafer uniformity problems. Their analysis can be Vacuum used to validate assumption 4 above. Reaction System. The LPCVD reactor considered in the simulation model is shown in Figure 1. It is operated 3-Zone Resistance Heater at pressures of 0.1-1 Torr. The vertical close stacking of G a s Control the wafers allows for a large throughput while taking adSvslem vantage of the fact that gas diffusivities are high a t these low pressures. This allows good transport of gases into the (b) region between the wafers (the interwafer region) and Inter-wafer region An?ular region hence a good radial uniformity of deposition. The flow in I Radial the region between the wafer edges and the reactor wall (the annular region) is laminar at typical LPCVD condiReactan Axial tions. The reactor walls and wafer surfaces are hot so that radial temperature gradients are small. The nonuniformity of growth rates in the radial direction is thus minimized. 10 lL However, a disadvantage of this strategy is that there is a significant deposition of film on the reactor walls. Figure 1. LPCVD reactor: (a) main components; (b) the wafer configuration (adapted from Jensen and Graves (1983)). In this paper, we consider the deposition of polycrystalline silicon (poly-Si) by the decomposition of silane: operating conditions, including the use of temperature Si(s) + 2H2(g) SiH,(g) staging and multiple injection points. This is a common method for poly-Si deposition. The A few studies have suggested modifications to the essilane is usually diluted by large volumes of hydrogen tablished multiwafer LPCVD systems to improve wafer carrier gas. The latter is also a product of the reaction, throughput and quality. Jensen and Graves (1983) proand thus its presence does tend to inhibit the deposition posed that a recycle system be added to combine the to some extent. The model does not include secondary benefits of a CSTR (continuous stirred tank reactor) and gas-phase reactions, which form intermediate species, e.g., a conventional LPCVD reactor. Roenigk and Jensen SiHz and Si2H6. (1985) discussed a continuous moving boat LPCVD reactor The gas-solid reaction rate is given by the nonlinear to minimize film thickness and composition variations from expression wafer to wafer and within each wafer. Both papers utilized the results from a fundamental LPCVD model to illustrate the advantages of the proposed systems. Another configuration for a LPCVD reactor is the single-wafer vertical-flow reactor (Foster and Learn, 1986). Although good The rate expression is based on adsorption-desorption uniformity is obtained, an obvious disadvantage to this equilibrium at the substrate surface, with an additional system is that much of the feed reactant is unconverted. term representing H2 gas inhibition. The following rate Below we discuss the simulation model and the configconstants were found by regression of experimental reactor uration of the LPCVD system studied here. The model data (Roenigk and Jensen, 1985): kl = (1.6 f 0.4) x IO9 was employed in a sensitivity analysis to determine the exp(-18500/T) mol/m2/s/atm, kH = (0.6 f 0.3) X lo2 important parameters. Finally, we present results on opatm-1/2, and k , = (0.7 f 0.1) X lo5 atm-'. timizing the design and operating variables subject to Based on the five assumptions listed earlier and the various constraints. previous development by Jensen and Graves (1983), the transport equations are as follows (see Nomenclature LPCVD Modeling section): The first comprehensive model of a LPCVD reactor was interwafer region presented by Jensen and Graves (1983). The basic asA d sumptions used in this model are as follows: - -(rIVrl) = -2R (1)There are no radial temperature gradients since the r dr reactor walls and substrate are heated and slow reaction with boundary conditions rates imply small heats of reaction. (2) The axial temperature profiie is controlled by furnace (3) settings because the gas heat-up lengths are small and most heat transfer occurs by radiation a t LPCVD conditions. annular region (3) There is no axial variation of the gas-phase composition in the interwafer region between any two consecutive 2R cw,l wafers since the interwafer spacing is small. r,(l + a ) + -7 dz A (4) There is no radial variation of the gas-phase composition in the annular region because the annular region where 7,the effectiveness factor, is defined as is small and there is rapid diffusion at LPCVD conditions. (5) The gas phase is at steady state, which is justified 2JrwrR(r) dr by the fact that CVD growth processes are slow compared (5) 1 7 = to gas-phase dynamics. rw2RIr, Middleman and Yeckel(l986) found that the assumpLe., the ratio of the average rate of deposition on a wafer tion of no axial variation of the gas-phase composition in
-
1164
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1 .oo
Table I. Effect of Process Parameters on the Depositiona change in av axial axial rate rate, variation, av eff Darameter Darameter A/min A/min factor fl% 1.89 -0.21 inlet SiH, mol 0.0006 fract 7.72 -1.02 pressure fO.l Torr 0.0028 temp f2 K 9.10 2.43 -0.0026 f0.005 m/s 0.59 -1.29 0.0008 inlet flow
0.98
50 L
1
3 0
0.96
li 2
*Base case: 10% SiH4, 0.53 Torr, 880 K, 0.05 m/s. *The overall change in the parameter is positive (from - to +) and is measured in absolute terms.
-8 5 c
0.94
-.
a
to that a t its edge. N,, and N,, are the molar flux components of SiH4 and
I.
a
L.
0.92
9
dX;
N,, = CD -
dz The boundary conditions for the annular region are
By use of eq 1, the rate R at r, varies in the z direction and hence 17 is a function of z. The effectiveness factor represents radial uniformity of the deposition. When 77 = 1, the surface reaction is rate-controlling; when 17 < 1, diffusion resistance becomes important, just as in heterogeneous catalysis. The pressure drop within the reactor was considered negligible. The increase in volumetric flow due to reaction stoichiometry was treated using the volume fraction expansion t in conjunction with fractional conversion a to represent mole fractions as (1 - 00x10 x20 + 201x10 x1 = for SiH4 x2= for H2 1 + ta 1 + €a (8)
The linear flow velocity is then 0
T TO
= uo(1 + ea)-
(9)
The nonlinear ordinary differential equations were put in dimensionless form (Jensen and Graves, 1983) and solved for LY and 17 using orthogonal collocation within nested Newton-Raphson iterations. The film growth rate can be found once a is specified.
Sensitivity Studies for the Reactor The computer model was used to predict the influence of changes in various process and design parameters upon deposition rates and uniformities. This was done in order to find the relative sensitivities of different parameters and to select the variables to be used in the optimization of the reactor performance. The nominal operating point on which the analysis is performed was based on an actual reactor configuration reported by Rosler (1977). We verified that the model assumptions were valid for the system under consideration. The reactor was 0.625 m in length and 0.12 m in diameter. Wafers were 100 mm in diameter and spaced 4.8 mm apart. The reactor pressure was 0.53 Torr, and the temperature was fixed at 880 K (independent of axial position). Figure 2 shows the deposition profiles for the base case. To perform the sensitivity analysis, a single parameter was varied while keeping all other constants at the above values. Table I presents the change in the average axial
6.90 0.00
0.20
0.40
0.60
0.80
1.00
Wrfrr dlsm. rrlrtlvr to rrrctor dlrm. Figure 2. Wafer growth rate profile and effectiveness factor for standard operating conditions (no optimization).
Table 11. Effect of Design Variables on the Deposition change in axial rate av axial variation, variable variable rate, A/min A/min av eff factor wafer spacing i10% 0.52 -1.13 0.0024 wafer radius *lo% -2.55 5.36 -0.0062
deposition rate (in angstroms/meter) and the average effectiveness factor (measured at the center of the reactor). Also presented is the change in axial variation of the deposition rate (i.e., the change in the difference between the rate at the inlet and that a t the outlet). Table I1 shows the effects of changing design variables on the deposition. Note that a negative change in the axial variation (signifying better axial uniformity of deposition) and a positive change in the effectiveness factor (signifying better radial uniformity) are the desired effects in improving the process performance. The most sensitive parameter affecting growth rate is temperature, due to surface kinetics being rate-controlling. As will be seen later, temperature is a good selection for the manipulated variable in the process optimization. However, this high level of sensitivity is a drawback because very precise temperature control in the reactor may be necessary. In addition, the increase in temperature has a relatively large adverse effect on axial and radial uniformities. Wafer radius mainly affects the axial variation in the growth rate. This design variable was not optimized because the radius must be the same for all the processes involved in the fabrication sequence. However, the effect of increasing the wafer radius (the reactor diameter is held constant) shown in Figure 3 is of considerable interest in view of the current trend toward using larger size wafers in the existing reactors. The radial and axial uniformities are seen to decrease sharply for wafer diameters larger than half the reactor diameter. The 8-in. wafers now being introduced in the microelectronics industry would apparently need a 16-in. reactor tube to achieve satisfactory uniformity of deposition. In that case, it may be desirable to consider alternative reactor configurations for the larger wafers.
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1165
0.2
0.0
0.4
0.6
0.8
1 .o
Normrllrrd reactor length
(b) n aa
0.0
0.2
0.4
0.6
0.a
1 .o
Normrllzod r r r d o r knglh
Figure 3. Deposition uniformity as a function of wafer to reactor diameter ratio.
The remaining parameters in approximate order of influence on the deposition uniformity are pressure, inlet flow velocity, interwafer spacing, and inlet mole fraction of silane. For the range of processing conditions under consideration, axial deposition nonuniformities are more severe than radial ones. The effectiveness factor (see eq 5 ) can be expressed alternately as 1=
average rate of deposition on wafer rate a t wafer edge
At an axial rate of 100 A/min, a change of 0.01 in 1 would represent only a 1A/min change in the average rate on the wafer. Hence, while the effectiveness factor should be included in any objective function for the optimization of the process, the primary concern is with axial uniformity. Optimization of the Reactor The two principal control variables that can be used to optimize the reactor performance are reactor temperature and reactant injection. Commercial deposition systems use a three-zone furnace to maintain different temperatures within the reactor. A second profile variable for improving the uniformity of deposition in these systems is to use reactant injection at different points along the reactor length, providing sudden increases in reactant concentration and hence deposition rate. Increasing the inlet flow rate reduces the drop in deposition rates along the reactor but at the same time decreases the conversion of the reactant. Although the objective function for such an optimization calculation should include cost factors representing the importance of reactant conversion relative to deposition rates and uniformities (which may be sitespecific), we do not consider the cost of unconverted reactants.
Interwafer spacing can also be used for optimization, but it is a design rather than a control (or operating) variable. It is very easy to obtain different spacings within the reactor, but Setalvad (1987) found that optimizing the spacing alone was ineffective in maintaining uniformity for a given reactor throughput. Based on these preliminary conclusions, several optimization problems using different combinations of design and control variables were investigated. The generalized reduced gradient method (Lasdon and Waren, 1983) was used for our studies, but other techniques, such as successive quadratic programming (Edgar and Himmelblau, 1988), would be satisfactory. The details of the numerical solution for this problem have been given by Setalvad (1987). Consider the nominal reactor design presented earlier in Figure 2 and Tables I and 11. While the radial uniformity is adequate, the axial deposition variation is about 10% of the rate a t the inlet. This is not surprising considering the large number of wafers (130) per batch. However, typical uniformities required in practice are less than 5 % . Therefore, any strategy to improve the performance of such a reactor should meet the 5% deposition uniformity criterion while maintaining or, if possible, increasing the wafer throughput. The general optimization problem is to find that set of values for the variables that maximizes the objective function (a measure of reactor performance) while satisfying the constraints (product specifications regarding uniformity and the bounds on the variable values). Temperature Profile as an Optimization Variable. As mentioned earlier, the growth rate is most sensitive to temperature. It is common practice in industry to adjust the temperature in three furnace zones along the reactor to improve the deposition uniformity. However, a rigorous procedure to optimize the zone temperatures is not normally performed. Using temperature as an optimization variable, we have applied an optimization code to determine the optimal temperature profile, similar to the well-studied tubular reactor temperature profile problem. It is important to note that too high a temperature change along the length of the reactor leads to a rate much higher than that at the inlet. While this increases throughput, it may be undesirable since the primary concern is the uniformity of the deposition. Further complicationsresult from the fact that within each temperature zone (i.e,, a length of reactor with constant temperature) the growth rate will decrease. In industrial applications, the temperature in each zone is determined by a trial-and-error procedure involving experimentation or empirical models. Such procedures usually cannot find the true optimum point and are inefficient and expensive, requiring extensive testing. The computer model incorporating rate constants obtained from actual reactor data was used in conjunction with an optimization routine to obtain the best possible set of temperatures. The predicted optimal temperature profile is subject to the accuracy of the model and the assumption that the temperature can be controlled in a piecewise, constant fashion. For this case, the variables to be optimized were Ti,i - 1, ..., n,,, where n, is the number of the temperature zones to be used. The optimization procedure was initiated with all Tivalues equal to 880 K. The objective function to be maximized was throughput, which can be expressed mathematically as
1166 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 120
(a)
1
E h
s
888
-
886
-
884
a
882
.
I
0.2
0.0
I
.
I
,
880
886.3*K
884.3.K
882.3'1:
1
bn.1
I
hn.2
* " " ' . "
a n 3
1.o
0.4 0.6 0.8 Normrllud roac101 pooltlon
Normrlhu reactor pornbn
Figure 5. Optimum temperature profile for three stages. 0.99
I
I
@timum valuex injection 1 vol = 0.5 W S x =012
inlet vo = 0.05 Ws
x 1 -0.1
"
injection 2 v = 0.a304ms 02
P
1
= 0'14
I '\ 1 I I I I I I I I I I1I I1 1 I I I I II II I I I I I1 I I I v
0.06 0.0
a
'
.
.
0.2
0.4
I
I
0.6
0.8
1.o
Nonnallzod roador porftlon Figure 4. Reactor performance with and without optimized temperature staging.
G is the growth rate in angstroms/minute, T i s the set of temperatures, and zi (i = 1,n) are points along the reactor at which the model is solved to obtain the rates Gi. This function is a weighted sum of the deposition rates over the entire reactor ( N is the number of increments). The objective function was maximized subject to the following constraints: (1) The maximum allowable axial variation in the rate is 5% of the maximum rate, max Gi - min G i V= I0.05 (11) max Gi (2) At no point is the radial variation in growth rate to be greater than 5%; in terms of effectiveness factors, qi 2 0.95
i = 1, ..., n,
(3) The temperature is restricted to 880 K ITiI 8 9 0 K i = 1, ..., n,
t-
0.05rrVS
01
.
(12)
(13)
This last constraint is imposed so that the grain size and other temperature-dependent material properties of the grown film and also its step coverage do not show excessive variations. The performance of the reactor with optimum threezone temperature staging can be compared to that of the original reactor (Figures 4 and 5 ) . The optimization routine increased the temperature of zone 3 the most, followed by zone 2. This strategy increased the value of f ( T) while decreasing the maximum axial rate variation (0.11 initially). The temperatures were increased from the initial guesses until the axial rate variation between the
x
11
I
v
t
02
0.1
x 12
= 0.05W~
-
0.1
Figure 6. Configuration and operating conditions for optimized reactant injection.
beginning and the end of zone 3 reached the 5% limit. Reactant depletion caused a sharp drop-off in rate within each zone. This effect increases noticeably from zone 1 to zone 3 (Figure 4a). The temperature in zone 2 could be decreased so that less reactant is consumed in this zone and more is available for zone 3. However, the resulting lower rates in zone 2 would cause the axial rate variation between the end of this zone and the beginning of zone 3 to exceed the 5% limit. Increasing the number of temperature stages can solve the problem of excessive rate drop-off within a zone. However, the resulting smaller temperature differences required between zones would be more difficult to maintain. Note that, at the low pressures and high temperatures of the LPCVD process, most of the heat transfer between the furnace and the substrate is expected to occur by radiation. Also, gas temperatures are relatively unimportant because the majority of the temperature sensitivity of the deposition results from the change in the solid-gas rate constant kl in eq 1. The effectiveness of temperature staging then depends mainly on the ability of the furnace to maintain the required temperatures within the reactor. Optimum Reactant Injection. An alternative to the use of temperature staging is to provide a sudden increase in the partial pressure of SiH,. This can be achieved by injecting additional reactant into the reactor at different points along its length. Assuming that there is little backmixing, sudden increases in growth rates at the injection points result, just as in the case of temperature staging, without the disadvantage of excessive rate drop-off due to reactant depletion. A schematic diagram for this configuration is illustrated in Figure 6.
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1167 The model was modified to incorporate the reactant injection as shown in Figure 6. The original reactor with two reactant injection ports can be considered, for modeling purposes, to consist of three smaller reactors or subreactors. Predicting the performance of the overall reactor then involves consecutively solving the modeling equations for each of the subreactors. The required inlet conditions for subreactor 1, i.e., gas flow velocity and silane mole fraction, are the same as those for the overall reactor. Inlet conditions for subreactor 2 are found by a mass balance based on the exit gas flow velocity for subreactor 1: uo’ = u,(l + €CY’) (14)
with injoctpn
c
where CY’is the conversion a t the exit of subreactor 1. The inlet gas flow velocity for subreactor 2 is then uo(2)
= uo’ + uo1
0.000
(15)
I
I
I
0.200
0.400
0.600
Roaetor Iongth (m)
where uol is the flow velocity of injection 1 (assumed to be at the same temperature as the exit gas from subreactor 1). The inlet mole fraction of silane for subreactor 2 is
(b)
with inioction
b
where xll is the silane mole fraction in injection 1and xld is the silane mole fraction in the gas exiting from subreactor 1, which is given by W
A similar mass balance involving the exit gas of subreactor 2 and injection flow 2 can be performed to yield the inlet gas flow velocity and silane mole fraction for subreactor 3. The model can now be solved for each of the three subreactors to yield the performance of the overall reactor. For this optimization case, the variables were the gas flow velocities uoi, i = 1, ..., ninj,representing the total amount of gas injected, and xli, i = 1, ..., ninj,the mole fraction of the reactant silane in each injection. Here nhj is the number of injection points. Two injection points were considered, giving four variables to be optimized. This was thought to be a reasonable balance between improved reactor performance and the resulting greater design and operational complexity. The objective function to be maximized was essentially the same as before except the rate G was now a function of uoi and xli instead of Ti, i.e.,
The uniformity constraints (eq 11-13) were again used in this optimization. Additionally, the bounds on the variables were 0.0 I xli 5 1.0 i = 1, ..., ninj (19) 0.03 Iuoi 5 0.5
i = 1, ..., ninj
(20)
The optimum values of the variables are shown in Figure 6, and the corresponding reactor performance is shown in Figure 7. As expected, the optimization code modified uol because the deposition was more sensitive to flow velocities (see Table I). After uol reached its upper bound, xll was increased until the axial uniformity constraint was violated; i.e., the difference in the rate between the end of the first zone and the beginning of the second was more
0.97 0.000
I
0.200
I
0.400 Rokrctor kngth (m)
I
0.600
Figure 7. Reactor performance with optimum staged injection.
than 5% of the inlet rate (Figure 7a). This was not true for injection point 2. Here the rates were very close to showing a 5% variation because not much drop-off occurred between the injection points. The small increase in the rate required was more easily brought about by increasing x12. The effectiveness factors (Figure 7b), unlike those in the temperature optimization, almost identically followed the axial rate trend. Interwafer Spacing as an Optimization Variable. The spacing between the wafers in the LPCVD reactor is also an attractive variable to optimize. To change the wafer spacing a t the design stage presents relatively few problems regarding set-up complexity. All that is needed is to machine a large number of slots in the wafer boat. Some slots then are left vacant to vary the spacing. While some work has been done where a single value of the interwafer spacing was optimized (i.e., wafers were spaced equally throughout the reactor), no system optimization using different spacings within the reactor seems to have been reported. Here the reactor was split up into zones of equal length, each of which had a different interwafer spacing. Technically, there could be as many zones as there are wafers. However, considering the trade-off between better deposition uniformity and the excessive computational effort that would be required for a large number of zones, the number of zones was set a t 10. The optimization variables chosen are the spacings in each zone, A,, i = 1, ..., n, (n, = number of wafer zones = 10). Because the length of the reactor is fixed, changing the interwafer spacings also changes the number of wafers
1168 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
I
I
I
I
I
~4.-0
01
02
03
I
I
I
I
I
I
I
04
05
06
07
08
09
10
\
wilh 130 unilormly 8plCsd wafers
Figure 8. Schematic diagram of the reactor configuration with optimum wafer spacing (numbers of wafers not to scale). 0.2
0.0
the reactor processes per batch. For this reason, a different objective function was used for this case: nn
f(A)=
C [Nw,i(Ai)12
i=l
(21)
1 .o
0.4 0.6 0.8 Normrlizrd rosetor posltlon
(b) 1.oo
f
I
wilh optimum rpuingr (1O'Zon.S)
where the number of wafers in the ith zone (of length
Lln,,) is
cwwd W
L
NW,,
= -
~ W S
0.98 with 130 uniionty
(22)
rpncbd w d r r s
n W Z 4
In other words, we maximized the total number of wafers the reactor of fixed length L can process per batch. The same uniformity constraints as before were used. The bounds on the variable were 2.1 mm I At I50 mm (23) where 2.1 mm (about 80 mils) is the lower limit on what is physically realizable. Velander and White (1987) have shown that the assumption used here, namely, diffusion is unaffected by the flow in the annular region, becomes invalid at an interwafer spacing of 50 mm of the wafer radius). The results of the optimization calculations are presented in Figures 8 and 9. These figures indicate that, in order to bring the reactor performance up to uniformity specifications, the number of wafers had to be reduced to 75 from the original 130 (without staging of temperature or reactant injection). The routine utilized the availability of different wafer spacings to redistribute the wafers so that as many as possible were placed nearer to the inlet where the reactant concentration was higher. The spacings in the first two zones were in fact a t the lower limit. It is conceivable that at higher pressures (and therefore lower gas diffusivities) the 0.95 effectiveness factor limit would prove to be the limiting constraint rather than simply an inactive bound on the variable. As the reactant was depleted further along the reactor, the interwafer spacing increased to the upper limit so that the rate a t the outlet was 5% lower than a t the inlet. Effectiveness factor values decreased the regions of constant spacing and increased in regions of increasing spacing. Also presented in Figure 9 are the results of having 75 uniformly spaced wafers in the reactor. This shows that there is some benefit in redistributing the wafers to prevent excessive rate drop-off. However, in an industrial situation, there would be little benefit in this redistribution if the process were now more sensitive to parameter changes. In order to investigate this possibility, the effects of a 20% decrease in gas flow rate and a 2 "C decrease in temperature (with a flat temperature profile and no reactant injections in both cases) on the performance of the reactor were examined. In both cases, the change in radial deposition uniformity was noticeably larger for the reactor with uniformly spaced wafers. This was also the case for the flow rate variation. The rate at the outlet changed to
0.0
0.2
0.4 06 0.8 Normrllrrd rrrctor porlllon
1.0
Figure 9. Reactor performance with optimum wafer spacings.
1.91 A/min compared to 1.63 A/min for the reactor with nonuniform wafer spacing. For the temperature variation, the reactor with uniformly spaced wafers exhibited a 4.42 A/min change in outlet rate. This was only slightly less than the corresponding 4.49 A/min rate change for the nonuniform wafer spacing case. Therefore, changing interwafer spacing alone is not in this case enough to bring the deposition up to uniformity specifications while still maintaining the same wafer processing throughput. However, these specifications can be met provided that the number of wafers processed per batch is decreased. Furthermore, there is apparently some benefit in redistributing the wafers as described above, resulting in less axial rate nonuniformity and a smaller sensitivity to flow rate variations. The sensitivity to temperature variations was not much different than for the uniformly spaced wafer case. The results indicated that it may be sufficient to use a reactor of one-third its original length since most of the deposition occurs in the first third of the optimized reactor. Interwafer Spacing Optimization in Conjunction with Temperature Staging. Earlier results indicated that, if some way is found to increase the reaction rates a t the low reactant concentrations toward the end of the reactor, the deposition could be brought up to uniformity specifications while maintaining, or perhaps increasing, the wafer processing throughput of the reactor. Temperature staging used together with the wafer spacing optimization can improve not only the uniformity of the deposition but also the number of wafers processed per batch. Thirteen variables, viz.,
Ti Ai
i = 1, ..., n,, i = 1, ..., nwz
n,, = 3 nwz= 10
were used in this optimization procedure. The procedure was initiated with all temperatures equal to 880 K and all spacing equal to 4.8 mm. The objective function used was the same as that in the wafer spacing optimization, representing the wafer pro-
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1169 (a)
The Optimized Temperature Staging
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Figure 10. Optimum wafer spacing and temperature profile.
cessing throughput of the reactor. Also, the same uniformity constraints (eq 11)and variable bounds (eq 13,19, 20, and 23) were used. The optimization results are shown in Figures 10 and 11. These plots demonstrate a significant improvement in the performance of the reactor. The deposition now has at most a 5% variation. At the same time, 31 additional wafers can be processed, spaced as shown in Figure lob. The 5% variation again occurred within the last temperature zone where reactant depletion and hydrogen inhibition were the largest. There is even more depletion than in zone 3 of Figure 4 because of the larger number of wafers processed prior to the last zone in this case. The variation is kept within the 5% limit by using a larger interwafer spacing (5.4 mm compared to 4.8 mm) in conjunction with a higher temperature.
Discussion of Results In comparing the optimization strategies, the following factors need to be kept in mind: (a) the axial and radial rates and uniformities achieved, (b) the relative ease with which the optimal values can be obtained and maintained in an actual reactor setup, and (c) the relative amount of computational effort required (depends mainly on the number of variables). Optimization of temperature staging produced fairly good results with respect to the axial rate. Reactant depletion, however, caused increasingly sharp drop-offs in the rate along the reactor. The depletion also led to decreases in radial uniformity (Figure 4b). On the positive side, this implies that relatively little reactant exists unconverted from the reactor. Most commercial reactor setups already have three-zone furnaces, but maintaining small temperature differences between the zones may present a problem. There were three variables to be optimized here, and the number of function calls required (i.e., the number of times the model was needed to be solved, representing computational difficulty) was 173. The reactant injection optimization produced better axial and radial rates and uniformities than those achieved by the temperature staging, due to the large amounts of reactant present. This, unfortunately, means that more reactant exists unconverted at the outlet of the reactor. Some modification of the reactor setup in Figure l a will
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Figure 11. Reactor performance with temperature staging and optimum wafer spacings.
be required' t o iliiorporate such injections. Many commercial reactors already have this capability, some with inlet ports all along the reactor. Reactant injections can be carried out using these ports. The optimization method for this case utilized four variables, and the number of function calls required was 346. This represented a greater amount of computation time than required above because each call involves solving the model three times, due to the multiple stages. Optimizing the interwafer spacings in conjunction with temperature staging produced excellent results, much better than optimizing either the spacings or the temperatures alone. Not only was the deposition brought up to uniformity specifications but the wafer processing capacity of the reactor was significantly increased. The use of 13 variables increased the number of function calls to 515. This still, however, does not represent an inordinately large amount of computation.
Summary and Conclusions Based on a LPCVD simulation model, several optimization strategies were devised to improve the deposition rates and wafer processing capacity of the horizontal, stacked-wafer reactor subject to uniformity constraints. The model was used in conjunction with a generalized reduced gradient algorithm to find optimum temperature staging and reactant injection schemes for the reactor. The use of three temperature stages and two reactant injections
1170 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
was beneficial. We also evaluated nonuniform interwafer spacing in the reactor to improve deposition uniformity and, together with temperature staging, increase the reactor throughput. Interwafer spacings were smaller toward the reactant inlet to take advantage of the larger reactant concentration in this region and larger at the end of the reactor, where reactant depletion and hydrogen production inhibited the poly-Si deposition. This scheme exhibited decreased sensitivity of the process to gas flow rate variations when compared to the uniformly spaced wafer case. Model predictions showed a sharp decrease in deposition uniformities above the wafer-to-reactor diameter ratio of about 0.5. This suggests that it may not be wise to use existing reactors for increased wafer sizes. Further, the reactor tubes that would then be necessary may have to be quite large and, in view of the low pressures, quite thick. The realization of the improvements projected from this simulation study will depend upon the accuracy of the model and the validity of the model assumptions. Experimental implementation will be carried out in a subsequent phase of this research. Of particular concern is the actual stepwise control of temperature in the reactor, which is affected by axial heat conduction in the furnace tube, and the degree of backmixing that occurs with reactant injection. Both of these effects will tend to reduce somewhat the benefits predicted in this paper. Acknowledgment This work was supported by a grant from the Texas Advanced Technology Research Program (TATRP). I.T. acknowledges the support of Texas Instruments. Nomenclature a = area of wafer boat relative to reactor tube area c = concentration co = total inlet concentration D = diffusivity f = objective function in optimization G = growth rate, A/min k = reaction rate constant L = reactor length N = number of points in summation N,, N , = molar flux N , = number of wafers n = number as denoted by subscript P = pressure p = partial pressure R = reaction rate r = radial coordinate r, = wafer radius
rt = reactor tube radius T = temperature V = maximum fractional axial variation u =
linear flow velocity
x = mole fraction z =
axial position
Greek Symbols = fractional conversion A = interwafer spacing, mm t = volume fraction expansion coefficient (Y
7 =
radial effectiveness factor
Subscripts
0 = inlet flow conditions; base conditions for D and T 1 = silane flow rates and concentrations 2 = hydrogen flow rates and concentrations b = bulk gas H = hydrogen inhibition rate constant inj = injections r = radial dimension s = rate constant in SiH, adsorption-desorption expression t z = temperature zones w = wafers wz = wafer zones z = axial dimension R e g i s t r y No. Si, 7440-21-3; SiH,, 7803-62-5.
Literature Cited Edgar, T. F.; Himmelblau, D. M. Optimization of Chemical Processes; McGraw-Hill: New York, 1988. Foster, D. W.; Learn, A. J. Deposition Properties of Silicon Films Formed from Silane in a Vertical-Flow Reactor. J. Vac. Sci. Technol. B 1986, 4(5), 1182. Jensen, K. F. Microreaction Engineering Application of Reaction Engineering to Processing of Electronic and Photonic Materials. Chem. Eng. Sci. 1987, 42(5), 923. Jensen, K. F.; Graves, D. B. Modeling and Analysis of Low Pressure CVD Reactors. J . Electrochem. SOC. 1983, 130, 1950. Lasdon, L. S.; Waren, A. D. Large-scale Nonlinear Programming. Comp. Chem. Eng. 1983, 7, 595. Middleman, S.; Yeckel, A. J. A Model of the Effect of Diffusion and Convection on the Rate and Uniformity of Deposition in a CVD Reactor. Electrochem. SOC.1986, 133(9), 1951. Roenigk, K. F.; Jensen, K. F. Analysis of Multicomponent LPCVD Processes. J . Electrochem. Soc. 1985, 132, 448. Rosler, R. S. Low Pressure CVD Production Processes for Poly, Nitride, and Oxide. Solid State Technol. 1977, 20(4), 63. Setalvad, T. M.S. Thesis, The University of Texas a t Austin, 1987. Velander, W.; White, D. Induced Convection Effect on IntrawaferUniformity in LPCVD. J . Electrochem. SOC. 1987, 134(4), 951. Received for review August 18, 1988 Revised manuscript received April 20, 1989 Accepted May 1, 1989
