IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 5 . NO. 3, AUGUST 1992
`
`16%
`
`Modeling of a High Throughput Hot-Wall Reactor for
`Selective Epitaxial Growth of Silicon
`
`Carl Galewski, Member, IEEE, and William G. Oldham, Fellow, IEEE
`
`Abstract-A tubular hot-wall silicon epitaxial reactor oper-
`ated in the selective deposition regime is characterized for
`growth rate uniformity in both the radial and longitudinal di-
`rections. The range of experimental conditions includes tem-
`peratures from 900°C to 8OO0C, pressures from 1 torr to 0.4
`torr, concentrations of SiH2CI2 in H2 from 17% to 4%, and
`wafer diameters from 125 mm to 75 mm. The simplest possible
`models that accurately predict these data are formulated; in the
`radial direction a simple first-order model is sufficient, whereas
`in the longitudinal direction it is necessary to include the en-
`trance region of the reactor and transport by diffusion. The
`resulting simulator is used to demonstrate improvements to the
`existing hot-wall reactor, and to propose a design for a scaled
`up production-sized hot-wall reactor. The hypothetical produc-
`tion-sized reactor accommodates 100 wafers of 200 mm diam-
`eter. Predicted throughputs for selective epitaxial silicon layers
`are 16 wafers/hour for 1 pm thick layers, and 27 wafers/hour
`for 1700 A thick layers.
`
`I. INTRODUCTION
`ANY element and useful device structures become
`
`M possible with the availability of selectively depos-
`
`ited epitaxial silicon layers. However, a low-cost ap-
`proach may be necessary to make them practical solutions
`for the complex integrated circuits of the future. An ex-
`ample of such a structure is the raised source/drain tran-
`sistor described ky Rodder and Yeakley [l] which re-
`quires a 1700 A
`thick selectively deposited silicon
`epitaxial layer. For this application, a tubular hot-wall re-
`actor able to deposit selective epitaxy at a rate of about
`100 A/min on a batch of 100 wafers would provide more
`accurate thickness control and lower cost than current
`cold-wall reactors, especially for wafer sizes of 200 mm
`and above.
`Typically, epitaxial deposition of silicon requires high
`temperatures that are not compatible with hot-wall reac-
`tors because of the severe depletion effects that result. We
`have found experimentally that it is possible with careful
`consideration of the reactor design, deposition condi-
`tions, and wafer cleaning to use low deposition tempera-
`tures that reduce depletion effects while still resulting in
`defect-free epitaxial silicon [2], [3]. Devices fabricated
`
`Manuscript received May 30, 1991; revised March IO, 1992. This work
`was supported by IBM Corporation and the Semiconductor Research Cor-
`poration.
`The authors are with the Department of Electrical Engineering and Com-
`puter Science, University of California at Berkeley. Cory Hall, Berkeley,
`CA 94720.
`IEEE Log Number 9201 185.
`
`on wafers with these epitaxial layers have been found to
`be indistinguishable from devices fabricated simultane-
`ously on standard substrates [4].
`It is the intent of this article to explore the feasibility
`of a production-sized hot-wall silicon epitaxy reactor for
`200 mm diameter. An experimental reactor has been con-
`structed by modifying a commercial low-pressure chem-
`ical vapor deposition (LPCVD) furnace [5], [6]. Growth
`rate data from this reactor is used to formulate a physical
`model that predicts the deposition uniformity in both the
`radial and longitudinal directions. The model is used to
`propose improvements to the existing reactor, and to pro-
`pose a design for a system that can accommodate 100
`wafers of 200 mm diameter. The throughput of the pro-
`posed reactor is found to compare favorably to a typical
`cold-wall reactor.
`
`11. MODELING OF HOT-WALL TUBULAR DEPOSITION
`SYSTEMS
`Low-pressure tubular hot-wall type reactors are pres-
`ently used to deposit many different materials, providing
`considerable impetus for improving the understanding and
`control of such reactors. Physical reactor models for
`polysilicon [7]-[12],
`silicon nitride [ 131, and silicon
`dioxide from tetraethoxysilane (TEOS) [ 141 have, for ex-
`ample, been desribed in the literature. None of these stud-
`ies, however, has included the actual entrance region of
`the reactor as a part of the model. The models published
`by Joshi [9] and Yeckel et al. [ l l ] include an entrance
`region, but only as an empty tube of constant temperature.
`The actual entrance region is quite complicated as shown
`in the schematic of our system in Fig. 1. Besides a rapidly
`changing temperature there is also a set of “baffles” that
`are used to reduce radiant heat losses. These baffles re-
`strict the flow and provide considerable surface area for
`deposition. Significant errors can arise if the ends of the
`reactor are ignored by applying ad-hoc boundary condi-
`tions at the ends of the region containing the load of wa-
`fers. In our model we will extend the simulation far
`enough toward the ends of the reactor so that boundary
`conditions of no deposition outside of the region are ac-
`curate.
`Our objective will be to construct the simplest possible
`models that explain our data. The physical processes that
`will be considered are: convection, diffusion, and chem-
`ical reaction. As illustrated by the arrows in Fig. 1, reac-
`
`0894-6507/92$03.00 0 1992 IEEE
`
`Page 1 of 11
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`Samsung Exhibit 1007
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`

`170
`
`IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 5, NO. 3, AUGUST 1992
`
`furnace
`tube
`\
`
`wafers
`/
`
`heat
`baffles
`/
`
`reactor wall
`
`vacuum
`Pump
`
`t
`gas in’et
`resistance heater
`Fig. 1. Schematic of the experimental hot-wall reactor. The arrows illus-
`trate the path of the gas inside the reactor.
`
`/
`
`tants are transported along the axis of the tube from left
`to right by convection and diffusion. The reaction that
`forms solid silicon on the wafers and reactor walls de-
`pletes the gas phase of reactants and generates reaction
`products. Transport by diffusion is a particularly impor-
`tant component of the reactor model because convection
`cannot efficiently aid the transport of gas to the center of
`the wafers which are oriented perpendicularly to the main
`gas flow. In most cases the resulting radial uniformity
`constraint determines the design and operation of the
`whole system. For example, reduced pressure operation
`is often necessary since it simultaneously decreases the
`deposition rate and increases the diffusion rate.
`111. RADIAL DEPLETION MODEL
`Yeckel et al. 1151 have shown that-for
`the typical
`LPCVD conditions encountered in a tubular hot-wall sys-
`tem-the dominant transport mechanism between the wa-
`fers is diffusion, and that the longitudinal concentration
`gradient across the wafer space can be ignored. In our
`model it will be assumed that a uniform concentration in
`the annular region supplies the gaseous reactant for each
`wafer, and that the reaction rate is first-order in SiH2C12.
`These simplifying assumptions can be justified by noting
`that our interest is in uniform conditions, implying small
`concentration variations. It also is assumed that the wa-
`fers are placed concentrically within the tube without the
`use of any cantilevers or “boats” to hold them in place.
`This is, of course, impossible in reality; however, it was
`found experimentally that the obstruction provided by the
`boat and cantilever improves uniformity. This observa-
`tion agrees with the improvement in radial uniformity de-
`scribed by Yeckel et al. [16] with wafer camers that de-
`liberately reduce
`the growth
`rate along the edge.
`Excluding the effect of the cantilever and boat is, there-
`fore, a worst-case condition.
`As shown in Fig. 2, the above assumptions reduce the
`radial depletion model to a simple one-dimensional mass-
`balance. In cylindrical coordinates the mass-balance can
`be written as
`
`where 6 is the space between the wafers, C is the concen-
`tration of SiH2C12, D is the binary diffusion constant of
`
`“I 1 I
`
`i + l
`i - 1
`i
`k S + 8 4
`
`wafers
`
`Fig. 2. Radial depletion model resulting from the simplifying assumption
`that transport occurs by radial diffusion from a uniform annular concentra-
`tion of reactant.
`
`SiH2C12 and H2, k,’ is the surface-rate constant for depo-
`sition, and a is the silicon coverage factor. The silicon
`coverage factor 01 is equal to the total area of silicon on
`the two wafers that bound the space divided by the total
`wafer area. It is included because deposition is selective
`with respect to S O 2 . If no oxide is present on either of
`the two wafer surfaces a = 2.
`The binary diffusion constant D [cm2/sec] for SiH2C12
`and H2 can be estimated from Chapman-Enskog kinetic
`theory [ 171 according to the following expression
`
`It is assumed in (2) that the ideal gas law is valid. The
`variables are defined as follows: P is the pressure [atm],
`T is the absolute temperature [OK], M is the respective
`molecular weights [ g / ~ o l e ] , is a characteristic diame-
`ter of the molecules [A], and !JD,AB is a slowly varying
`function of the dimensionless temperature kT/E. A more
`convenient representation for our purposes is given by
`
`(3)
`
`where Do at temperature To and pressure Po is calculated
`according to (2). The overall temperature dependence of
`(2) is approximated by the power 1.65 in the temperatures
`range of interest [ 171. If Lennard-Jones parameters for
`CH2C12 are used as an analog to SiH2C12 in (2) then Do is
`5700 cm2/sec at the typical operating conditions of 850°C
`and 0.6 torr.
`The first-order deposition rate expression is of the form
`
`where PDcs is the SiH2C12 partial pressure, and k, is a
`reaction-rate constant obeying an Arrhenius relationship
`with activation energy EA. Conversion of the deposition
`rate to a flux as a function of concentration is accom-
`plished by defining a new surface-rate constant kl as in
`Rdep = k,’ C = ksmsiRTC [moles/sec-cm2]
`(5)
`
`Page 2 of 11
`
`

`

`GALEWSKI AND OLDHAM: MODELING OF HIGH THROUGHPUT HOT-WALL REACTOR
`
`.-
`
`where mSi is the molar density of silicon equal to 8.3 1
`moles/cm3, R is the ideal gas constant, Tis the ab-
`solute temperature, and C is the concentration of SiH2C12
`[moles/cm3].
`The concentration Ci is constant in the annular region
`of the ith wafer. At the center of each wafer the cylindri-
`cal symmetry of the problem forces the gradient to zero.
`The result is the following boundary conditions for (1):
`(6)
`C(R,) = Cj
`
`(0) = 0
`
`(7)
`
`dr
`A general solution to (1) can be obtained by defining
`the following dimensionless parameters:
`C
`$ E - ci
`r
`4 E -
`RW
`
`(9)
`
`I
`
`The dimensionless factor + is analogous to the Thiele pa-
`rameter used in chemical engineering. After substituting
`(8), (9), and (10) into (1) we obtain the following dimen-
`sionless differential equation
`
`with boundary conditions:
`
`d* - (0) = 0.
`d4:
`The solution for the normalized concentration $ defined
`by (1 1) is a Bessel function of zero-order and parameter
`t . The normalized concentration at the center, $(+, 0),
`can be obtained by a series approximation. Neglecting
`high-order terms, an expression accurate to 1 % for radial
`depletion up to 20% is given by
`
`Thus it is possible to predict the uniformity once the ratio
`ki / D is known. Because of the first-order rate assumption
`there is no dependence on the concentration of SiH2C12 in
`the annular region, making the radial uniformity model
`independent of longitudinal depletion.
`A series of epitaxial silicon depositions were performed
`on ( 100) oriented substrates to obtain a model for ki / D .
`The experimental conditions are summarized in Table I.
`The temperature, pressure, input gas composition, and
`wafer diameter were varied around a nominally standard
`set of conditions represented by experiments 177 and 182.
`
`TABLE I
`RADIAL UNIFORMITY EXPERIMENTS
`DCS Wafer dia.
`cmm)
`(sccm)
`
`Press.
`(mtorr) Hz (sccm)
`
`Temp. ("C)
`
`852
`852
`852
`85 1
`85 1
`85 1
`90 1
`803
`852
`852
`
`620
`622
`622
`62 1
`393
`1003
`607
`596
`60 1
`605
`
`400
`400
`400
`400
`200
`800
`400
`400
`312
`452
`
`32
`32
`32
`32
`16
`64
`32
`32
`64
`16
`
`100
`100
`75
`125
`100
`100
`100
`100
`100
`100
`
`Expt. #
`
`~
`
`177
`182
`183
`184
`186
`188
`191
`192
`193
`194
`
`Wafers with a 1 pm thick oxide were patterned with nar-
`row concentric oxide circles spaced 4 mm apart in order
`to maximize depletion and to maintain radial symmetry.
`The whole-wafer pattern was printed on a transparency
`and transferred to the wafers by contact photolithography.
`The wafers were etched in 5 : 1 buffered HF after a 20 sec
`flood exposure and development. The resulting oxide pat-
`tern consisted of 200 pm wide oxide lines, corresponding
`to 95% silicon coverage independent of wafer size. The
`wafers were placed in an open boat with slots cut at 2.4
`mm intervals so that data for spacings of 2.4, 4.8, 7.2,
`and 9.6 mm could be obtained. Backs of the wafers were
`bare, resulting in a worst-case oxide coverage ratio QI of
`1.95.
`Film thicknesses were measured with a stylus profiler
`after removal of the oxide lines with HF. The uncertainty
`of the thickness measurements was estimated by measur-
`ing a designated reference sample before, and after, each
`use of the stylus profiler. From thts reference sample we
`obtained a 3a variation of +70 A when averaging the
`readings from 4 consecutive scans for each measuremmt.
`The actual thickness measurements spanned 8,080 A to
`6,090 A, resulting in an uncertainty between + 1.6% and
`f2.2% for normalized data.
`Experiment 177 in Table I was an initial exploratory
`experiment that differed from experiment 182 only in that
`two sets of variably spaced wafers were used. This ex-
`periment demonstrated that the effect of longitudinal de-
`pletion was less than the thickness measurement accu-
`racy, validating the choice of a first-order approximation
`in (4). Raw deposition rate data versus radial position from
`experiment 177 are shown for the upstream set of wafers
`in Fig. 3(a). The radial position axis corresponds to a left
`(negative) to right (positive) diameter as viewed facing
`the wafer from an upstream position inside the reactor.
`The left-to-right diameter is well away from the effects
`caused by the boat and cantilever. It is difficult to discern
`a trend from the data in the raw form because of the effect
`of depletion. If, however, the growth rate is normalized
`to the growth rate at the left (negative) radial position, the
`influence of spacing and oxide selectivity becomes clearer
`as illustrated by Fig. 3(b). The normalized growth rate
`corresponds directly to the normalized concentration II,
`because of the first-order reaction-rate assumed in (4).
`
`Page 3 of 11
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`

`

`172
`
`52
`
`.
`
`SPACINGS : - 2.4 mm, Oxldb back
`-
`
`IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 5 , NO. 3, AUGUST 1992
`
`1
`
`.
`
`
`
`1
`
`.
`
`
`
`r
`
`.
`
`
`
`lowing expression
`
`2.4
`
`I DIAMETER - io0 mA
`
`42
`-60
`
`I
`
`-40
`
`0
`
`-20
`(a)
`
`.
`
`'
`20
`
`
`.
` '
`40 60
`
`SPACINGS :
`
`- 2.4 mm, oxlda back
`- 4.8
`
`1 EXPT. 177
`
`DIAMETER = 100 mm
`-40
`-60
`0
`-20
`
`1
`
`20
`
`40
`
`60
`
`1 1 .
`
`
`
`'"I .
`
`. . . . . .
`.
`SPACINGS : 0 2.4 mm, oxide back
`* 2.4
`, 4.8
`
`'
`
`'
`
`'
`
`'DIAMETER - 100 mm
`
`EXPT. 177
`
`(15)
`The 1.93 eV activation energy of (15) agrees well with
`the range of activation energies of 1.3 to 2.2 eV that have
`been reported in the literature for deposition of silicon
`from SiH2C12. The final model obtained by substituting
`(15) into (14) accurately represents the radial variation
`across the wafers as shown by the solid lines in Fig. 3(c).
`For design purposes, the most important aspect is the
`maximum difference in thickness between the edge and
`middle of each wafer. In Fig. 4 the normalized rates
`measured at the centers of the wafers are plotted as points
`with error bars versus the wafer spacing with wafer di-
`ameter, pressure, temperature, and composition as pa-
`rameters. The values calculated after substitution of (15)
`into (14) are shown as the solid lines in Fig. 4. The model
`is within the error bars of the measured data except for a
`few cases involving the narrowest 2.4 mm spacing. There
`is some variation in the actual spacing caused by the slots
`in the boat having to be wider than the wafers they hold.
`The resulting wafer movements affect the results obtained
`from the narrowest spacings more than the results from
`wider ones. Also an important consideration concerning
`the narrowest spacing is that the mean free path is 0.5 mm
`at 850°C and 0.6 torr. As a result, the growth rate at the
`edge may be larger than predicted by (14) for narrow
`spacings approaching the mean free path.
`An effectiveness factor 9 can be used to incorporate the
`effect of radial depletion in the longitudinal deposition rate
`model that will be described in the next section. The for-
`mal definition of the effectiveness factor is the ratio of the
`actual deposition rate across the whole wafer to the rate
`evaluated using the concentration at the edge of the wafer.
`The definition of this effectiveness factor is analogous to
`those used in chemical engineering to relate reaction rates
`in catalyst pellets to external concentrations. For the lin-
`ear reaction rate expression defined shown in (5) the
`expression for the effectiveness factor becomes
`Rw
`ki C(r) 27rr dr
`9 = kJ C(R,) 7rRk
`
`(16)
`
`'
`
`An expression for k J / D was obtained by dividing ki as
`defined by (4) and (5) with the diffusivity D as represented
`by (3). The result is an equation with two fitting param-
`eters, a constant multiplier and an activation energy. The
`values of these parameter were obtained by minimizing
`the sum of square errors between growth-rate data and
`calculated values for the normalized concentration ac-
`cording to (14). Best fit to the data resulted from the fol-
`
`Substituting the definition for the normalized variables
`defined by (8) and (9), and integrating over the low-order
`terms in the series approximation of the Bessel function
`that satisfies (1 l), we obtain the following simple expres-
`sion
`
`+2
`1 + -
`8
`+2 .
`1 + -
`4
`Because of the first-order reaction rate, the effectiveness
`
`9=-
`
`Page 4 of 11
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`

`

`GALEWSKI AND OLDHAM: MODELING OF HIGH THROUGHPUT HOT-WALL REACTOR
`
`1'3
`
`1
`
`0.9
`
`0.8
`
`0.7
`0
`
`5
`
`10
`
`15
`
`20
`
`(a)
`
`r i l l . . . . . . . . # .
`
`A 620mton
`0 1OOOmtorr
`.
`
` . . . . . . . I
`
`1
`
`0.9
`
`0.8
`
`0.7
`
`1
`
`0.9
`
`0.8
`
`0.7
`0
`
`5
`10
`15
`WAFER SPACING (mm)
`(d)
`Fig. 4. Comparison between measured data (points) and radial depletion
`model predictions (solid lines) for experiments that vary: (a) the nominal
`reactor temperature, (b) the total pressure, (c) the concentration of SiH2CI,
`in the input gas stream, and (d) the wafer diameter.
`
`20
`
`factor in (17) is independent of the variations in reactant
`concentration along the length of the reactor. The effec-
`tiveness factor concept is also valid for more complicated
`reaction rate expressions, but would no longer be inde-
`pendent of reactant concentrations. In this case, the effec-
`tiveness factor would have to be included explicitly in the
`formulation of the longitudinal model.
`
`IV . LONGITUDINAL UNIFORMITY MODEL
`The non-dimensional Reynolds and Peclet numbers
`characterize the flow regime in the reactor. The transition
`from laminar to turbulent flow occurs for Reynolds num-
`bers exceeding 1000, whereas a straightforward evalua-
`tion for our reactors predicts values between 1 and 4 [6].
`A Reynolds number larger than 0.1 characterizes a tran-
`sition away from creep flow around objects in the path of
`the gas. The flow is, therefore, expected to be laminar,
`but with vortices forming behind edges of objects in the
`reactor. For simplicity the effect of vortices will not be
`included.
`The Peclet number is a measure of the relative impor-
`tance of convection versus diffusion. It is defined by
`U L
`Pe - D
`in which U is the linear velocity, L a characteristic length,
`and D the diffusion coefficient. If the Peclet number is
`much greater than 1, convection dominates. If the Peclet
`number is much less than 1, diffusion dominates. The
`same reactor conditions used for the Reynolds number
`calculation result in Peclet numbers between 1 and 3 [6].
`Both convective and diffusive transport must, therefore,
`be included in the longitudinal model.
`Because of the low pressure and high temperature, it
`will be assumed that radiation is the dominant heat trans-
`fer mechanism, and that heat of reaction does not affect
`the temperature profile. The reactor temperature will,
`therefore, be radially uniform and equal to the wall tem-
`perature. Furthermore, for typical operating conditions,
`the radial diffusion rate is about lo3 times larger than the
`growth rate, leading to the assumption that reactant con-
`centration in the annular region is constant in the radial
`direction.
`Since temperature and reactant concentration vary sig-
`nificantly in the longitudinal direction, a simple first-or-
`der reaction-rate expression, such as (4), is not expected
`to be a good approximation. The thermodynamics of the
`Si-C1-H system and the kinetics of silicon deposition from
`chlorosilanes have been extensively researched, but there
`is no consensus on a reaction mechanism. The large num-
`ber of gaseous species that can be formed, and the signif-
`icant reverse component due to etching by HC1, makes it
`unlikely that a single reaction mechanism is responsible.
`Even for a simple heterogeneous reaction, different and
`quite complex reaction-rate expressions result from as-
`suming that the rate limiting step is: gas phase reaction,
`adsorption, surface reaction, or desorption [ 181. To avoid
`the ambiguity resulting from postulating a reaction mech-
`anism, an empirical power-law rate-expression describing
`the overall chemical reaction will be used.
`The overall chemical reaction that describes the depo-
`sition of silicon in our system can be written as
`SiH2C12 c--) Si,,, + 2 HC1
`H*
`Hydrogen is not consumed or generated by this reaction
`
`Page 5 of 11
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`

`

`I14
`
`IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 5, NO. 3. AUGUST 1992
`
`but can affect both the forward and reverse rates so that a
`general deposition rate expression in units of cm/sec is
`rdep = k s ~ g , ~ & - k r ~ ~ c / ~ d H ,
`(19)
`where the two reaction rate constants are given by
`k, = k,,e-E"/kT
`(20)
`kr = k
`e - E o r / k T
`(21)
`ru
`It is not expected that the powers will be integers or half-
`integers because this expression does not describe a single
`mechanism. All 8 reaction parameters (k,,, E,, k,,, E,,,
`a , b, c , d ) will, therefore, be varied to best fit the depo-
`sition rate data.
`Evaluation of (19) along the reactor requires calcula-
`tion of the concentration profiles for SiH2C12 and HC1.
`It will be assumed that HCl is in equilibrium with the
`SiH2C12 at each point in the reactor (or equivalently that
`their respective transports are not significantly different).
`The HC1 concentration can then be calculated from the
`amount of SiH2Cl2 that has reacted. The formulation of
`the model is simpler if the amount of reacted SiH2C12 is
`expressed by a conversion factor x defined as
`C
`
`x = 1 - -
`Cnoreuc
`where C is the concentration of SiH2C12 and CnUre,, is a
`concentration profile calculated by setting the reaction rate
`to zero. The HCI concentration along the reactor is related
`to CnUrea, according to the following relationship
`(23)
`cHC/ = 2XCnoreac
`where the factor 2 results from the stoichiometry of the
`overall reaction.
`After applying the above assumptions, the model re-
`duces to a one-dimensional single-species mass-balance
`similar to one discussed by Levenspiel [19] for a plug-
`flow reactor with longitudinal dispersion due to diffusion.
`The differential equation describing the reactor is derived
`by dividing the reactor into N elements. For each element
`a steady-state mass-balance is written so that the mass-
`flow entering it is equal to the mass-flow leaving. For the
`ith element, shown in Fig. 5 , mass enters the element by
`convection, diffusion, and by the possible use of an injec-
`tor at that location. Mass leaves the element by convec-
`tion, diffusion, and deposition. Each element has the
`properties of length Az, temperature T, surface area per
`unit length SL, and cross-sectional area X,. Also associ-
`ated with each reactor element is the concentration C of
`reactant and the volumetric flow rate U [cm3/sec] due to
`the forced flow of gas through the reactor. In the limit of
`Az -, 0 the following differential equation is obtained
`
`(22)
`
`The first term represents convection due to the volumetric
`flow rate and the second term represents diffusion. The
`
`deDosition
`t
`
`convection
`i -
`diffusion
`
`t
`
`injection
`
`convection
`
`diffusion
`
`&(i) = length
`Tli) = temoerature
`
`SL(i) = surface areafunit length
`XAIi) = cross-sectional area
`
`Fig. 5 . The mass-balance for the ith reactor element in the finite difference
`formulation of the longitudinal depletion model. Each element has the
`properties of length, temperature, surface area per unit length, and cross-
`sectional area.
`
`(25)
`
`last term represents the net change in reactant due to de-
`position and injection. Note that both Rdep and R, are
`expressed in terms of a molar flux [moles/sec-cm2].
`The boundary conditions for (24) are
`C(0) = c,
`dC
`- (L) = 0
`dz
`where the reactor length from 0 to L is chosen so that
`deposition outside the region can be ignored. The concen-
`tration C, in (25) is the input concentration of SiHzClz at
`the front of the reactor. Equation (26) results from the
`assumption that no reaction takes place beyond the dis-
`tance L (i.e., C cannot change past L ) , which was shown
`by Wehner and Wilhelm [20] to be the correct boundary
`condition for Peclet numbers greater than zero.
`For the case of a first-order reaction-rate expression,
`constant temperature, and no volume expansion, (24) can
`be solved analytically for C as a function of distance z
`[20]. However, in our case the reaction rate is not first-
`order, and the extremes of the reactor, where the temper-
`ature varies rapidly are included. A numerical solution of
`(24) for C ( z ) is instead necessary. A finite difference ap-
`proximation of the differential equation the result in a tri-
`diagonal set of nonlinear equations that can be solved with
`the Newton-Raphson algorithm.
`All terms that are necessary to solve (24) numerically
`have been discussed previously except for the volumetric
`flow rate. It arises because of the volume of gas that is
`forced through the system by the vacuum pump. The pres-
`sure drop can be estimated from the analysis performed
`by Hitchman et al. for LPCVD of polysilicon [21]. From
`their analysis we estimate that at the typical reactor con-
`ditions of 850°C and 0.6 torr the pressure drop is 14 mtorr
`over the length of the hot-zone. Because of the small mag-
`nitude of the drop, we will assume that the pressure is
`constant, forcing instead the volumetric flow rate to vary
`in order to compensate for volume changes due to tem-
`perature and reaction. According to this assumption the
`
`Page 6 of 11
`
`

`

`GALEWSKI AND OLDHAM: MODELING OF HIGH THROUGHPUT HOT-WALL REACTOR
`
`TABLE I1
`LONGITUDINAL UNIFORMITY EXPERIMENTS
`
`Expt. #
`195
`196
`197
`198
`199
`200
`20 1
`
`Temp. ("C)
`90 1
`853
`852
`853
`852
`853
`804
`
`Press.
`(mtorr)
`609
`604
`387
`986
`604
`602
`602
`
`H2 (sccm)
`400
`400
`200
`800
`452
`312
`400
`
`1-5
`
`DCS
`(sccm)
`32
`32
`16
`64
`16
`64
`32
`
`volumetric flow-rate becomes
`
`is the input volumetric flow rate at standard
`where v,,d
`conditions (1 atm and 273"K), x is the previously defined
`conversion factor, and f is the volumetric expansion factor
`(i.e., the amount of gaseous volume change resulting from
`complete conversion of all of the reactants to product).
`Seven epitaxial silicon deposition experiments on
`(100) oriented substrates were conducted to find best fit
`values for the reaction rate parameters in (19). The de-
`position conditions used for the longitudinal uniformity
`experiments are shown in Table 11. Temperature, pres-
`sure, and composition were varied around the nominally
`standard deposition conditions represented by experiment
`196. Wafers with a 1 pm thick oxide were patterned by
`standard photolithography. The openings in the oxide
`mask were created by etching in 5 : 1 BHF. The epitaxial
`thickness was measured by a stylus profiler as described
`in the radial model section. Six patterned wafers were in-
`cluded in each run, and placed inside the reactor at the
`locations shown in Fig. 6. To obtain deposition-rate data
`from the front of the reactor, two wafers were placed di-
`rectly on the cantilever rods, and one wafer was placed
`between the last set of front baflles. The remaining 23
`wafers were blank dummy wafers to maximize depletion.
`Deposition is selective with respect to the oxide on
`wafers, but the quartzware accumulates a coating of sili-
`con after repeated exposures. Inspection of the quartz-
`ware inside the reactor reveals an abrupt beginning and
`end of the deposition zone. Using a coordinate system
`with the center of the reactor at zero, the locations where
`deposition starts and ends are approximately -37 and
`+37 cm, respectively. Simulation is confined to this re-
`gion by applying the boundary conditions of (25) and (26)
`to the observed deposition boundaries. A grid of 120 uni-
`formly spaced points along the reaction zone was found
`to produce a smooth deposition profile even through re-
`gions with abrupt transitions. (The simulator can, if nec-
`essary, accommodate a non-uniform grid to increase the
`number of points in regions with abrupt transitions.) Tem-
`perature, surface area per unit length, and cross-sectional
`area that described the reactor during the experiments were
`input to the simulator as a function of z as illustrated in
`Fig. 7. The temperature profile in Fig. 7(a) was measured
`with a movable thermocouple at 2.5 cm intervals. The
`accuracy of the temperature profile is estimated to be k0.3
`cm and f0.4"C. Interpolation was used to obtain tem-
`peratures for grid points between temperature measure-
`ment points. The profiles for surface area per unit length
`and cross-sectional area were calculated from the dimen-
`sions and locations of the various pieces of quartzware
`inside the furnace. The surface area per unit length profile
`shown in Fig. 7(b) is calculated from the actual surface
`areas before radial depletion was incorporated for the dif-
`ferent reactor conditions by multiplying radial surface
`
`oxide patterned
`/ wafers \
`
`boat for
`125 rnm wafers
`
`\
`/
`- - +
`rear heat-baffles
`front heat-baffles
`0
`Fig. 6. Locations of the oxide patterned wafers used to measure the growth-
`rate profiles for the longitudinal depletion experiments.
`
`950 1
`
`700'
`-50
`
`400
`
`I
`
`.
`
`I
`50
`
`1st BOAT m
`
`.
`
`'
`-25
`
`. " '
`25
`(a)
`
`0 1
`-50
`
`.
`
`'
`-25
`
`0
`
`(b)
`
`25
`
`I
`50
`
`FRONT BAFFLES
`REAR BAFFLES
`0 -50
`-25
`25
`50
`0
`DISTANCE FROM CENTER (cm)
`(C)
`Fig. 7. Profiles for (a) temperature, (b) surface area per unit length, and
`(c) cross-sectional area that describe the hot-wall epitaxial reactor during
`the longitudinal uniformity experiments.
`
`Page 7 of 11
`
`

`

`I76
`
`IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING. VOL. 5 . NO. 3, AUGUST 1992
`
`I
`
`---- SIMULATION
`
`17% DCS
`
`200,
`
`I
`
`OL
`
`-50
`
`1
`
`150
`
`F 2 100
`E
`2 50
`
`0
`
`TABLE 111
`LONGITUDINAL MODEL PARAMETERS
`
`4.930 . 10'
`2.037
`4.922 . io4
`I .763
`0.583
`0.551
`1. I29
`-0.097
`
`areas with the effectivness factor in (17). The large re-
`duction in cross-sectional area at each end in Fig. 7(c) is
`caused by the baffles that are placed inside the reactor at
`each end to reduce the heat losses from the center of the
`reactor.
`A downhill simplex algorithm was used to minimize the
`sum of squared errors between simulated deposition rates
`and the data. The values for the reaction rate parameters
`obtained by this method are shown in Table 111. From our
`previous discussion on the complexity of the real reaction
`mechanism it would be dangerous to infer too much phys-
`ical significance from these parameters. It is reassuring,
`nevertheless, that the activation energies in the longitu-
`dinal model are within the range of those reported in the
`literature. However, the approximately half-order de-
`pendence on H2 is surprising. It is commonly believed
`that H2 mainly acts to reduce the deposition rate by oc-
`cupying surface sites, which should result in an inverse