1950 J. Electrochem. Soc.: SOLID-STATE SCIENCE AND TECHNOLOGY September 1983 3. These observations are temporarily interpreted in terms of PGMA cross-linking variation basically de- pendent on the ionization quantity of air caused by electrons ejected from the gold film fabricated on the x-ray mask. 4. In order to suppress electrons ejected from this gold film, a few protective films have been examined. It appears that silicon nitride and Mylar films satisfy the demand. Acknowledgment The authors wish to thank Drs. S. Asanabe, K. Ayaki, D. Shinoda, and Y. Okuto for their continuous en- couragement. Manuscript submitted Dec. 27, 1982; revised manu- script received April 21, 1983. NEC Corporation assisted in meeting publication costs oy this article. REFERENCES i. J. R. Maldonado, G. A. Coquin, D. Maydan, and S. Somekh, J. Vac. Sci. Technol., 12, 1329 (1975). 2. Y. Saitoh, H. Yoshihara, and I. Watanabe, Jpn. J. Appl. Phys., 21, L52 (1982). 3. J. M. Moran and G. N. Taylor, J. Vac. Sci. Technol., 16, 2020 (1979). 4. K. Okada, Jpn. J. Appl. Phys., 20, L22 (1981). 5. D. Maydan, G. A. Coquin, J. R. Maldonado, S. Somekh, D. Y. Lou, and G. N. Taylor, IEEE Trans. Electron Devices, ed-22, 429 (1975). 6. K. Suzuki and J. Matsui, J. Vac. Sci. Technol., 20, 191 (1982). 7. H. J. Fitting, Phys. Status Solidi A, 26, 525 (1974). 8. See, for example, H. A. Bethe, "Handbuch der Physik," Vol. 34, S. Flugge, Editor, p. 62, Springer- Verlag, Berlin (1958). 9. See, for example, L. V. Spencer and U. Fano, ibid., p. 121. Modeling and Analysis of Low Pressure CVD Reactors K. F. Jensen and D. B. Graves Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455 ABSTRACT A detailed mathematical model for the hot wall multiple-disk-in-tube LPCVD reactor is developed by using reaction engineering concepts. This model includes the convective and diffusive mass transport in the annular flow region formed by the reactor wall and the edges of the wafers as well as the surface reactions on the reactor wall. In addition, the model describes the coupling of diffusion between and reaction on the wafers. Variations in gas velocities and diffusion fluxes due to net changes in the number of mols in the deposition are also taken into account as are nonisothermal operating condi- tions. The combined reactor equations are solved by orthogonal collocation. The deposition of polycrystalline Si from Sill4 is considered as a specific example, and the model is employed in estimation of kinetic rate constants from published reac- tor measurements. The effects on the growth rates and film thickness uniformity (within each wafer and from wafer to wafer) of variations in flow rates, reactor temperature profiles, and Sill4 concentration in the feed stream are analyzed. The model predictions show good quantitative agreement with published experimental data from different sources. Finally, recycle of reactor effluent is considered a typical commercial operating conditions, and it is demonstrated that this modification produces higher growth rates and better film uniformity than can be achieved in conventional LPCVD processing. Chemical vapor deposition (CVD) is a key process in the semiconductor industry for growing thin solid films, in particular polycrystalline St, Si3N4, and SiO2 films. These materials are grown predominantly in tubular, hot wall, low pressure chemical vapor deposition (LPCVD) reactors which have replaced the earlier cold wall atmospheric pressure reactors. This change has come about because LPCVD reac- tors allow a larger number of wafers to be processed with better uniformity of film thickness and composi- tion within and among the wafers. Table I gives a summary of LPCVD process conditions for polycrys- talline St, Si3N4, and SiO2. The very large packing densities of wafers that can be realized in LPCVD re- actors are po:ssible because at the low process pres- sures (0.5-1 Tort), the diffusion coefficients are three orders of magnitude larger than at atmospheric pres- sure so that the chemical reactions at the surface of the wafers are rate controlling rather than mass transfer processes. Moreover, in spite of the low pressures, rates of deposition in LPCVD are only five to ten times less than those obtained in atmo- spheric CVD since the reactants are used with little or no diluent in LPCVD whereas they are strongly diluted in conventional CVD (1). The wall deposits, which are avoided in cold wall reactors, could create particulate problems in LPCVD processes. However, these potential problems may be avoided by keeping the cumulative wall deposit thin by periodic reactor cleaning. Key words: LPCVD, polycrystalline Si, modeling, reactors. The development of CVD reactors and the selection of operating regimes have hitherto mainly been based on empirical design rules. Tilts limits the operation of existing reactors to certain fixed conditions and se- verely hampers the development of novel deposition processes. Furthermore, the decrease of feature sizes in large scale integrated circuits necessitates the growth of films of highly uniform thickness and com- position. This must, of course, be achieved by efficient, economical use of resources. Thus, reaction engineer- ing analysis and design should be a key element in the development and operation of CVD reactors. CVD in cold wall reactors at atmospheric pressure has been analyzed for two major reactor geometries. Eversteijn et al. (2), Rundle (3), and Takahashi et al. (4) used various simplifying asumptions to consider the deposition of Si in a horizontal reactor with a flat plate st~sceptor. Fujii et al. (5), Dittman (6), and Manke and Don'aghey (7) modeled to various degrees of accuracy the epitaxial growth of Si in a "barrel" Table I. Typical deposition conditions in LPCVD reactors (1), (21) Total Pres- ~as Reactant Temper- sure, flow, Dep. rate, Film gases ature, ~ Torr sccm A/min Poly-Si Sill, 620 0.3 70-100 i00-120 SiH,/N2 640 0.5 500-1000 140-160 SisN4 SiH~/NH~ 8i)0 0.3 100-200 40-60 SiO2 SiH2C1JN.~O 800-90,0 0.6 200-300 80-120
`
`[001]
`
`[002]
`
`[003]
`
` Ex.1009 p.1
`
`

`
`[004]
`
`[005]
`
`Vol. 130, No. 9 MODELING AND ANALYSIS 1951 react'or. None of the investigators took into account that the barrel-formed susceptor may rotate during reactor operation. There have been tew and mostly descriptive modeling efforts for LPCVD reactors. Gieske et al. (8) and Hitchman et al. (9) have pre- sented experimental data and discussed flow fields, ma~s transfer effects, and possible kinetics but have not formulated a detailed model. Recently, in fact after we had completed this work, Kuiper et al. (10) published a model for the LPCVD growth of polycrystalline Si. This simple model omits effects of diffusion in the space between wafers and the sig- nificant volume expansion associated with Sill4 re- acting to Si and 2H2. Furthermore, the model in its present form is restricted to isothermal conditions and therefore cannot predict reactor temperature profiles that will yield uniform growth rates throughout the reactor. The authors make no attempt to compare the model predictions with experimental observations. In this paper, we develop a detailed mathematical model for the commercial, multiple-disk-in-tube LPCVD reactor illustrated in Fig. 1. This model in- cludes the physicochemical processes in the annular flow region formed by the t~be wall and the edges of the wafers as well as those occurring in the spaces between the wafers. The changes in gas velocity and diffusion fluxes, due to a net production/reduction of mols in the gas phase during the reactions, are taken into account as is the deposition of material on the re- actor wall and wafer carrier. A general nonisothermal furnace profile is considered such that variations in growth rates from wafer to wafer may be minimized. We shall demonstrate that the mathematical model correctly predicts experimental observations and gives a quantitative comparison with published experi- mental data from separate reactor studies. In addition. we shall use the reactor analysis to compare growth rates and thickness uniformities across individual wafers as well as from wafer to wafer for different operating conditions. In particular, the effects of flow rates, diluent (N2.). temperature profiles, and recycle of reactor effluent are elucidated. Model Development The modeling approach that we present is not restricted to any specific deposition kinetics. How- ever, in order to compare model predictions and ex- periments, we focus on the low pressure CVD of polycrystalline Si from Sill4 which is a major LPCVD process for which experimental data have been pub- lished. We consider the usual commercial LPCVD re- Fig. I. Schematic of LPCVD reactor: (a) overall system, (b) de- tails of wafer positioning. actor illustrated in Fig. 1. The wafers are placed perpendicular to main direction of flow and concen- trically inside a quartz tube. The flow in the annular region formed by the reactor tube and the wafer edges, i.e., Rw --~ r ~ Rt, is laminar. Because of the low pressures, diffusion is the dominant mode of transport in the spaces between the wafers, and the mixing caused by the flow past the wafer edges is negligible. The re'actor temperature is controlled by a three-zone heater, and the heats of reaction associated with the deposition are small because of the slow growth rates. Moreover, at the low pressures and high temperatures encountered in LPCVD, heat transfer occurs mainly by radiation. Therefore, we may assume that the tempera- ture profile is .determined solely by the furnace set~ tings. Even though there have been a number of ex- perimental studies of Si deposition from Sill4, the kinetic mechanism is not understood (11). It is possi- ble that Sill4 decomposes in the gas phase (12, 13) according to the reaction Sill4 ~ Sill2 + H~ [ 1 ] The formed SiH2 then adsorbs on the Si surface Sill2 + * ~ Sill2* [2] where it diffuses to a reaction site and decomposes to Si releasing H2 Sill2*--> Si(s) + H2(g) + * [3] Since there are no rate data available for the in- dividual reaction steps, we assume, as did Claassen et al. (14), that the surface reaction is rate control- ling and follows the rate expression kPsiH4 = [mol Si/m 2 sec] [4] 1 -k K1PH2 q- K,2PsiH4 This particular form may be justified by the mecha- nism (Eq. [1]-[3]) and reflects the experimental ob- servations: (i) the rate is inhibited by H2, (ii) the rate is first order in Sill4 at low Sill4 partial pressures (~10 m Torr), and (iii) the rate approaches zero order in Sill4 at high partial pressures (Nt Torr) (14). In addition, we may neglect the homogeneous reactions (Eq. [1]) in the ensuing model development and the lumped rate expression (Eq. [4]) used. Considering the lack of detailed kinetic data, the inclusion of the kinetic scheme (Eq. [1]-[3]) in the model would only obscure the reactor analysis. With the above assumptions and remarks, we are able to construct mathematical models for the station- ary operation of the LPCVD reactor shown in Fig. 1. The concentration of the reactant Sill4 between the wafers is governed by the continuity equation V (cid:12)9 N1 = 0 [5] where N1 is the molar flux of Sill4 and P hrl = -- ~ DlmVXl -b xl ,~ hri [6] RT i=l Here D~ra is an effective binary diffusivity and x rep- resents the tool fraction of Sill4. The boundary con- ditions are ~Xl xl(r=Rw) =Xlb ~ =0 Zl<Z<Zl~h [6a] Or o --Nlzlwafer ! = ~ Nlzlwaferi+l = ~ 0 < r < Rw [6b] where expression [6b] states that the molar flux to the Wafer surfaces equals the rate of deposition of Si per unit area. Since the wafer spacing h is small compared to the radius of the wafer Rw, i.e., A/Rw < 0.1, and the reactant concentration varies smoothly over the length of the reactor, we may neglect the variation in the
`
`[006]
`
`[007]
`
`[008]
`
` Ex.1009 p.2
`
`

`
`[009]
`
`[010]
`
`1952 J. Electrochem. Soc.: SOLID-STATE SCIENCE AND TECHNOLOGY September 1983 z-direction within each cell formed by the wafers and use an average Sill4 concentration. By the averaging process, the boundary conditions (expression [6b]) are incorporated into the continuity balance which now takes the form a d (rND -- ~ [7] 2r dr By using Eq. [6], that 2N1 = --N2 according to the stoichiometry, and that the flux of inerts N3 -- 0, the balance takes the form P d( r dxl ) -- -- Dim = ~ [8] 2r RT -~r 1 + xl dr with boundary conditions xl(r=Rw) =Xlb and dxl =0 [9a, b] dr o The continuity balance over Sill4 in the annular flow region has the form V (cid:12)9 cDVxl -- V (cid:12)9 (cxtv) = 0 [10] D is a dispersion coefficient and D -- Dim since at low pressures and small Re the mixing occurs mainly by molecular diffusion. The boundary conditions are --D Oxl = vo[xjo -- xl] @z o 0Xl =0 .Oz L and P Oxl ----D RT Or [lla] Rw < r < Rt [11b3 -- ~(Rt) [llc] Rt O<z<b P D = ~ + a [lld] RT or Rw Equations [11a] and [11b] are the same as Danck- werts' boundary conditions for fixed bed reactor mod- els. With these boundary conditions, the dispersion model becomes identical to the continuously stirred tank reactor (CSTR) in the limit of very large dis- persion coefficients, and it approaches the plug flow re- actor (PFR) in the limit of very small dispersion coeffi- cients where convection dominates. The boundary condition (Eq. [llc]) represents the deposition on the hot reactor wall. The first term on the right-hand side of condition (Eq. [lld]) gives the flux of Sill4 be- tween the wafers due to the deposition of Si. The quantity ~ is analogous to the effectiveness factor in heterogeneous catalysis (16) and is defined as the ratio of the total rate of reaction on each pair of wafers to that which we would obtain if the concentration were equal to the bulk concentration everywhere in the cell formed by the two wafers, i.e. R w 2.~0__ r~(xl(r))dr ~q -- [12] Rw2~ (Xl (R~)) Thus, if the surface reaction is the rate controlling step, ~1 ~ 1; whereas if the diffusion between the wafer controls is the rate controlling step, n < 1. In the limit of strong diffusion resistance, the deposition is con- fined to a narrow outer band of the wafers and a strongly nonuniform film results. The second term on the right-hand side of Eq. [lld] refers to the deposi- tion on the wafer carrier..~ is the area of the carrier relative to reactor tube area. The pressure drop over the LPCVD reactor is very small (9) so that the total pressure, P, may be con- sidered constant. Therefore, the increase in the num- ber o~ mols by the deposition reaction SiH~ -> Si + 2H2 means an increase in the volumetric flow. To effectively treat this effect, we express the concentra- tion of species in terms of the overall conversion (x) and the fractional change in volume between no conversion and complete conversion (e) (see Ref. (20) for details). The stoichiometry implies that e ---- xl0. The mol fractions of Sill4 and H2 are then given by (1 -- x)xlo x~o + 2xx10 xl -- x2 = [13] 1 + r 1 + ex The gas flow is parallel to the axis of the reactor (the z-direction) and v = Vo(1 + ~x)T/To [14] where Vo is the entrance velocity. At the low pressures employed in LPCVD, diffusion coefficients are approximately 1000 times larger than those at atmospheric conditions and consequently, the radial mixing is rapid in the annular flow region. Furthermore, the time scale associated with deposition is larger than that for diffusion, i.e., the generalized DamkShler number, ~ (cid:12)9 (Rt -- Rw)/CoDm is small. Therefore, we assume perfect radial mixing and aver- age over the continuity balance by using ~Rtr.drdo 0 R w <'> = [15] ~2wf Rt rdrdo 0 =]R w This averaging eliminates the r-component and includes the boundary conditions Eq. [11c] and [lld] in the one-dimensional continuity balance. Then by making use of the expressions [13] for the tool fractions, the axial velocity (expression [14]), and the temperature dependence of the diffusivity D/Do = (T/To) 1.65, we derive the following balance in terms of the conver- sion, X D~ q--~z +V~ 2~ Rt(l+.~) + ~ ~ =0 /16] (Rt 2 -- Rw 2) C10 where -- (1 + ex)2 [17] The boundary Conditions take the form dx dx = vo(1 -{- (cid:141) -- 0 [181 D~ "~-z 0 "~z L To set forth the fundamental parameter combina- tions associated with the deposition process, we make the modeling equations dimensionless by defining g(x) = Z r ~=-- 4=-- L Rw 2LRt(1 -t- ~) Dal = ~(x = 0) Vo (Rt 2 -- Rw 2) ci0 2L Rw2/~ De2 = ~(x _-- 0) vo(Rt2 -- Rw2)cl0 ~(x) VoL Pe = ~(x = 0) Do 2Rw2~(x = 0) 4~2 = [19] Acl0Dlmo (T/To) 0.65 The reactor Eq. [16] then takes the form
`
`[011]
`
`[012]
`
`[013]
`
` Ex.1009 p.3
`
`

`
`[016]
`
`[014]
`
`[015]
`
`Vol. 130, No. 9 MODELING AND ANALYSIS 1953 d-")- ~P +Pe--d~---Pe(Da~-t-~lDa~)g(x) =0 [20] with boundary conditions dx I dx = Pe(1 q- x(0),)x(0) and ~ = 0 [21a, b] o and. f0 g(x(O) df [22] ~] = 2 ~ g(xb) The reaction-diffusion problem governing the wafers becomes d~ ]'+ex d~- +~g(x(O) =0 [231 with boundary conditions = 0 [24a, b] x(~_--l) --Xb(D and d~ o The parameters each have specific physical signifi- cances. The Damk6hler numbers Da~ and Da2 repre- sent the ratios of reactor space time to the character- istic time for deposition on the reactor wall and on the wafers, respectively. The axial Peclet number (Pc) represents the ratio of the time constant for con- vective transport to that for diffusive transport. The parameter, ~, is equivalent to the Thiele modulus (r used extensively in analysis of heterogeneous reac- tions. It denotes the ratio of the characteristic time for diffusion in between the wafers to the characteristic time for deposition of Si on the wafer surfaces. Thus, if ,I, is large, the deposition is hindered by diffusion and a nonuniform film results. This effect is com- pletely analogous to that of a large r for a porous catalyst. In fact, the modeling Eq. [20]-[24] are simi- lar to those for a catalytic fixed bed reactor. Numerical Solution of the Modeling Equations The modeling equations form two nonlinear bound- ary value problems which must be solved by some suitable numerical technique. Here we use the or- thogonal collocation method which has been applied successfully to many chemical reaction engineering problems, especially fixed bed reac,tors [ (17), (18) and references therein]. In this method, the first and sec- ond spatial derivatives are approximated by a weighted sum of values of the dependent variable at the col- location points N+I -- = Aijxj i = 0, 1 ..... N + 1 [25] d~ ~i j=o N+I d2x = Bijxj i = O, 1 .... , N -k 1 [26] d~ "2 ~i j=0 The weight matrices depend on the trial functions which in our case are the first N Jacobi polynomials Pi(a,/~)(~ ") with weight function ~a(1 -- ~)~ (17, ct. 3). Since there is no special symmetry in the tubular re- actor problem Eq. [20], we use (a, f~) = (0, 0) for that equation. However, for the wafer problem (Eq. [23]), advantage is taken of the radial symmetry and the Dirchlet condition (Eq. [24a]) by using (a, g) = (1, 0). The collocation points are in each case the zeros of the orthogonal polynomium, PN(a,B)(~). By dis- cretizing the reactor Eq. [20], [21], we obtain the fol- lowing set of nonlinear algebraic equations for the interior N+I N+I N+I ~i ~r [Pe-t- ~--~Aijfj ] ~ Aijxj j=o j=o j=o -- Pe(Dal --k ~(X)Dla~)g(xi) = 0 i --- 1,..., N [27] and for the boundary N+I ~Aojxj -- Pc(1 q- XoD Xo = 0 [28] j=0 N+I ~ AN+I,jXj = 0 [29] j=0 In order to solve this system of equations, we need to evaluate ~](xi) from the solution to the wafer problem, (Eq. [23]). Since the variations in film thickness across the individual wafers is small (<5%) in most LPCVD reactors, we expect r to be small. In that case, it is possible to obtain an ac- curate solution to the wafer problem by using just one interior collocation point. This technique has been demonstrated to give accurate results for the ana- logous cat'alyst particle problem even for moderately large values of (cid:12)9 (19). Then by using the one-point collocation technique (19), the wafer problem is ap- proximated as [ 1-k ~x(~i) ] ec2g(x (h)) ---- 6 In [30] 1 + ex(1) This equation can be solved for given r e, and x(1) to yield x(~l). One can then evaluate ~q~ by Radau quadrature (17, 19) as 2 ~]i-- -- [% " g(xQ1)) + 1/a g(x(1))] g(x(1)) 1 [ g(x(~l)) ] =--- 1-t-3-- [31] 4 g(x(1)) By combining these equations with the N q- 2 non- linear algebraic equations for the reactor tube (Eq. [27]-[29]), we can find the solution to the en- tire LPCVD problem by Newton-Raphson iteration. This, in turn, enables us to predict growth rates and variations in film thickness within and among wafers. The growth rate on the wall is determined by Gt ----- Vsi~( z ~-- 0)g(x) [32] while the average growth rate on each wafer is Gw ---- Vsi~(x =- O)g(x)~ [33] The average growth rate on all wafers in the reactor is computed by quadratures I: G = Vsilq (x = O) g(x)nd~ N - 2 -- Vsi~ (x -- 0) wig(xi)~li [34] i=l where the w~'s are Gauss quadrature weights corre- sponding to the N interior collocation points. The vari- ation in growth rates and thus film thickness is de- termined as ~0 ~ [g (x)n]Zdt = ~ d~= [ f:g(x)~ld~]--1 L~,U j [35a] for the reactor (cid:12)9 w 2 = 2 ~ -= d~ Gw 2 f01 [ g(x(O) ]2d~__l [35b ] r12 " g(x(1)) within each wafer
`
`Ex.1009 p.4
`
`

`
`[017]
`
`[018]
`
`[019]
`
`[020]
`
`50' J. Electroehem. Soc.: SOLID-STATE SCIENCE AND TECHNOLOGY September 1983 These integrals can also be evaluated by quadrature. With the developed mathematical model, we are now able to analyze experimental observations and to draw comparisons between growth rates and possible levels of uniformity at different operating conditions. Evaluation of Kinetic Parameters Since detailed rate data for the pyrolysis of SiHi are scarce and often contradictory (11), we chose ~to evaluate the parameters k, K1, and K~ in expression [4] from experimental data for the dep- osition of polycrystalline Si in an LPCVD reactor ob- tained by Claassen et al. (14). The reactor was loaded with 50 76.2 mm diam wafers spaced 10 mm apart and it was kept isothermal. Very low concentrations of SiHt (xl0 = 0.0047) were used in mixtures of H2 and N2. The relative amount of the carrier gases, H2 and N2, was varied to demonstrate the inhibition of Si deposition by H2. Figure 2 shows the experimental data from Claassen et al. (14) and the theoretical pre- dicted growth rates as a function of the dimensionless axial reactor position, ~. The model predicts quite well the decrease in growth rates along the reactor length due to deple- tion of Sill4 as well as the inhibiting effect of in- creased Hz concentration. The rate constants determined from this comparison with experimental data are k = 1.25 109 exp (-- 18,500/T) mol/m2/atm/sec K1 = 1.75 103 atm -1 K2 = 4.00 104 atm-1 The activation energy for k is not determined but in- stead extracted from Bryant's survey of the tempera- ture dependence of Si deposition (11). The "adsorp- tion parameters," K~, K2, are considered temperature independent in the range of temperatures of interest in LPCVD of polycrystalline Si, 600~176 because of insufficient data. The rate constants were not deter- rained by any statistical technique although the modeling equations could form the basis for non- linear least squares parameter estimation. However, that falls outside the scope of the present work. In order to test the ability of our mathematical model to predict growth rates at different conditions than those used in the evaluation of the kinetic param- eters, we consider the effect of varying the deposition area by removing wafers. Figure 3 shows mea- 30 25- 20- tS- tO- 0 ~ XH2 : 0 X XH2 : O.2S ~ (cid:12)9 XH2 = 0,995 S i } , I ' } ' I ' 0.0 8,2 0.4 8.6 0.8 I .0 REACTOR POSITION Fig. 2. Model predictions (solid line) vs. experimental data (14) of growth rate profiles as a function of inlet H2 concentration. Reactor conditions: P = 0.53 Torr, T = 6250C, flow = 1000 sccm, XSiH4 = 0.0047. 0 5 WAFERS 40- 30- 28- 1954 10 l ] l t ' 1 ' I ' 0.0 0.2 0.4 0.6 0.8 .0 REACTOR POSITION Fig. 3. Model predictions (solid line) vs. experimental data (14) of growth rate profiles as a function of number of wafers in the reactor. Reactor conditions: P = 0.53 Torr, T = 625~ flow 1000 sccm, Xs~H4 = 0.0047. sured growth rates by Claassen et al. (14) and pre- dicted growth rates for the cases of 5 and 50 wafers in a 0.hm reaction zone. The model matches the trend of the experimental observations and demonstrates that, as expected, the growth decreases more rapidly in the case of the larger deposition area (50 wafers) because of the faster depletion of the reactant, Sill4. The assumption of a small wafer spacing relative to the wafer radius (i.e., A/Rt << 1), which was used to eliminate the z-dependence in the wafer problem (Eq. [5], [6]), clearly does not hold for the five-wafer case. This may explain the slightly larger differences between model predictions and experimental observa- tions for the five-wafer run than those for the 50- wafer run. Comparison of Model Predictions and Commercial Process Data In order to further test the mathematical model, we consider deposition data obtained by Rosler (1) with a different LPCVD reactor system, and at SiH~ concentrations two orders of magnitude larger than those used in the above evaluatins of the kinetic constants. The same size wafer (76.2 mm diam) was used in the experimental study. The reactor contained 110 wafers spaced at 4.7 mm 'as well as five dummy wafers at each end of the wafer zone. The possible growth rates and film thickness uniformities for two typical conditions for commercial LPCVD reactors, pure Sill4 feed and Sill4 diluted with N2, were com- pared, and the effects of flow rates, reactor tempera- ture profile, and dilution were demonstrated experi- mentally. Figure 4 illustrates the measured growth rates (1) and the predicted rates as functions of the wafer position in the reactor for a (cid:127) change in SiH~ feed rates around the base case of 47 sccm pure SiH.~ feed. The reactor temperature is varied from 607 ~ to 654~ along the length of the reactor by using the 3-zone furnace to produce a nearly uniform thickness of the polycrystalline Si layer in the base case. The model predictions accurately predict the trends in the experimental data. When the feed rate of Sill4 is stepped up 15%, the growth rates, as reflected by the Si-thickness, increase along the length of the reactor since there is excess reactant available relative to the base case. A fiat thickness profile could clearly be achieved by lowering the temperature gradient along
`
`[021]
`
`[022]
`
` Ex.1009 p.5
`
`

`
`[023]
`
`[024]
`
`[025]
`
`VoL 130, No. 9 MODELING AND ANALYSIS 1955 the length of the reactor. Similarly, if the flow rate of SiI-I4 is dropped, the rates of deposition of Si tail off. Again a flat thickness profile could be reestablished by readjusting the temperature profile, this time by in- creasing the temperature gradient. The slight wavelike appearance of the model pre- dictions comes about because the deposition rate de- pends exponentially on the reactor temperature which varies piecewise linearly along the reactor. In order to obtain a quantitative agreement between model and experiments, we have used a slightly smaller tempera- ture variation, 607~176 rather than 607~176 and a shorter deposition time, 30 rain rather than 45 min. Nevertheless/the accurate prediction of the trends and the good quantitative agreement between model pre- dictions and experiments obtained with those minor changes demonstrates the usefulness of our modeling approach, in particular when considering that the ki- netic rate constants are based on deposition at very low Sill4 concentrations and on different substrates than in the present case. Because of the high sensitivity to SiI~ flow rate variations and the large temperature gradients re- quired to produce a flat growth profile in LPCVD processing with pure SiI-I4, dilution with N2 is often employed. This process modification reduces the sensi- tivity to flow variations and to temperature gradients as seen by comparing Fig. 4 and 5. The latter figure shows experimental data for dilute processing (23% Sill4 in N2), obtained by Rosler (1), and model simu- lations. Again the model predicts the trends quite well, but lower temperatures were used in the model simulations in order ~o compensate for our imperfect knowledge of the rate constants. The growth rates obtained by using pure Sill4 are less than those obtained in dilute processing, approxi- mately 125 A/rain vs. 200 A/min. However, this differ- ence is caused by the larger Sill4 flow rate used in dilute processing. For the same feed rates of SiH~, the growth rate is larger for pure Sill4 feed than for di- lute feed. Nevertheless, the dilute processin~ offers the advant,ages of a small sensitivity to flow disturb- ances and a small temperature gradient along the re- actor. The latter effect is especially advantageous since it produces more uniform grain sizes and con- sequently more uniform electronic properties than 0.6 (cid:127)I(cid:127) 0. 5 . o (cid:12)9 : 0,4" I,- 0.3- 0.2- (cid:12)9 1 (cid:12)9 X X 47 SCCM 40 SCCM I I ' I ' I ' I ' I i 20 40 60 80 100 120 Wafer Position Fig. 4. Model predictions (solid line) vs. experimental data (1) of film thickness profiles as a function of (pure) Sill4 flow rate. Reactor conditions P ~ 0.6 Tort, wafer spacing ---- 0.00476m, re- actor length ---- 0.625m (assumed); experimental temperature pro- file and deposition time: 607 ~ 629 ~ 654~ t ~ 45 min; model temperature profile and deposition time: 607 ~ 614 ~ 650~ t 30 mln. 0.60 ~ 0.55. -~ 8.58- m o a. 8.45. 8.48 SiH 4 X 170 SCCN (cid:12)9 15o sccM (cid:12)9 130 SCCM X X X "0~ A & k ~ I ' I J I , I , I' l 0 28 40 68 88 lSe 128 Wafer Position Fig. 5. Model predictions (solid line) vs. experimental data (1) of film thickness as a function of (dilute) Sill4 flow rate. Reactor conditions same as in Fig. 4 except: P ----- 0.7 Torr, N2 flow -- 500 sccm; experimental temperature profile and deposition time: 643 ~ 643 ~ 647~ 25 rain; model temperature profile and deposition conditions: 625 ~ 629 ~ 633~ 25 min. can be achieved in a process with pure Sill4 feed where larger temperature gradients are needed to give uniform film thickness. The disadvantages of the di- lute processing are the high flow rates and the asso- ciated pumping speeds. Furthermore, there is an in- creased cost of SiI4_4 since less Sill4 is converted over the reactor. Aa LPCVD Reactor with Recycle The ultimate goal in LPCVD deposition of poly- crystalline Si is to grow Si films with constant thick- ness and material properties. Because of the distrib- uted nature of the LPCVD reactor, it has been neces- sary to either vary the wafer temperature along the length of the reactor or to use dilute processing in order to at least approach this goal. Neither of the two approaches is ideal as dr in the above discussion. A continuously stirred tank reactor would give uniform films, but there are difficulties such as particulate and wafer loading problems associated with the design and operation of a CSTR reactor. On the other hand, these problems are avoided in the tubular LPCVD reactor. By recycling a fraction of the reactor effluent to the reactor inlet as illustrated in Fig. 6, one can combine the benefits of the CSTR and the conventional LPCVD reactor. If we define the re- cycle ratio, ~. as reactor effluent returned to the reactor inlet Fr total reactor effluent -- Fexit then the recycle reactor will behave as CSTR in the limit ~, --> 1 and obviously as a conventional LPCVD reactor in the limit ~ --> 0 [see Ref (20)]. Thus, the inlet 1 outlet recycle stream F r Fig. 6. LPCVD reactor schematic with recycle. Recycle ratio Fr/Fexit.
`
`[026]
`
` Ex.1009 p.6
`
`

`
`[027]
`
`[028]
`
`[029]
`
`1956 J. Electrochem. Soc.: SOLID-STATE SCIENCE AND TECHNOLOGY September 1983 recycling provides a means for obtaining various de- grees of mixing with the LPCVD reactor. If we assume that the recycle stream is quenched to avoid reactions in the recycle line and then mixed with the inlet stream at the same temperature, a total mol balance over the mixing point yields Fo -~ Fr : Fo' [36] where Fo, Fr, Fo' are the molar flow into the system, in the recycle loop, and into the reactor, respectively. By using Fr -- ~.Fo'[1 + e(xe -- xi)] we find the flow into the reactor is Fo Fo' = [37] 1 -- ~[1 -~ e(Xe-- xi)] where xi is the conversion of Sill4 in the combined feed to the reactor and Xe represents the exit conver- sion. A balance over Sill4 gives FOX10 of- FrXle --" Fo'Xlt [38] which by using Eq. [13] can be written in terms of the inlet and exit conversions, xi and (cid:141) as Fo ~,(1 -- Xe) [1 + ,E(xe -- xi)] (1 -- xi) -- [