Elements
`of Chemical
`Reaction
`
`Engineering
`
`Second Edition
`
`H. SCOTT FOGLER
`
`Ame and Catherine Vennema Professor
`
`of Chemical Engineering
`The University of Michigan, Ann Arbor
`
`LAM Exh 1008-pg 1
`
`

`
`Library of Congress Cataloging-in-Publication Data
`
`Fogler, H. Scott.
`
`Elements of chemical reaction engineering / F. Scott Fogler. -
`2nd ed.
`
`(Prentice-Hall international series in the physical
`-
`cm.
`p.
`and chemical engineering sciences)
`Includes bibliographical references and index.
`ISBN 0-13-263534-8
`1. Chemical reactors.
`TPlS7.F65
`1992
`
`II. Series.
`
`I. Title.
`
`660'.28-DC20
`
`91-15994
`CIP
`
`Editorial/production supervision: Jacqueline A. Martin
`Prepress buyer: Kelly Behr
`Manufacturing buyer: Susan Brunke
`Supplements editor: Alice Dworkin
`Acquisition editor: Michael Hays
`Editorial assistant: Dana L. Mercure
`
`
`
`© 1992 by Prentice Hall P T R
`A Simon & Schuster Company
`Englewood Cliffs, New Jersey 07632
`
`All rights reserved. No part of this book may be
`reproduced, in any form or by any means,
`without permission in writing from the publisher.
`
`Printed in the United States of America
`
`109
`
`ISBN [1-L3-El=353'+-B
`
`Prentice-Hall International (UK) Limited, London
`Prentice-Hall of Australia Pty. Limited, Sydney
`Prentice-Hall Canada Inc., Toronto
`Prentice-Hall Hispanoamericana, S.A., Mexico
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`Simon &. Schuster Asia Pte. Ltd., Singapore
`Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
`
`|
`
`1
`
`LAM Exh 1008-pg 2
`
`

`
`If an industrial
`ECHO“ is not mass
`zransfer-limited, it
`is probably run
`incorrectly.
`
`L D_ Schmidt. U.
`of Minn_
`
`TABLE ll-1
`
`Variation ofReaction Rate with:
`T-”_") 0f —’?_—‘?:"_’——m
`Limitation
`Velocity
`Particle Size
`Temperature
`
`External diffusion
`Internal diffusion
`Surface reaction
`
`U":
`Independent
`Independent
`
`(d,,)‘3'2
`(d,,)"'
`Independent
`
`zlsinear
`Exponential
`Exponential
`
`through the bed, particle diameter, and temperature for the three types of
`limitations we have been discussing.
`The exponential temperature dependence for internal diffusion limi-
`tations is usually not as strong a function of temperature as is the dependence
`for surface reaction limitations. If we would calculate an activation energy
`between 8 and 24 kJ/moi, chances are that the reaction is strongly diffusion-
`limited. An activation energy of 200 kJ/mol, however, indicates that the
`reaction is reaction rate—limited.
`
`11.8 Chemical Vapor Deposition (CVD)
`Reactors
`
`As discussed in Section 6.6, CVD is a very important process in the mi-
`croelectronics industry. The fabrication of microelectronic devices may in-
`clude as few as 30 or as many as 200 individual steps to produce chips with
`up to 106 transducers per chip. An abbreviated schematic of the steps in-
`volved in producing a typical computer chip is shown in Figure 11-11.
`Starting from the upper left we see that single crystal silicon ingots are
`grown in a Czochralski crystalizer, then sliced into wafers, and chemically
`and physically polished. These polished wafers serve as a starting material
`for a variety of microelectronic devices. A typical fabrication sequence is
`shown for processing the wafer beginning with the formation of an SiO2
`layer on top of the silicon. The S103 layer may be formed either by oxidizing
`a silicon layer or by laying down a SiO2 layer by Chemical Vapor Deposition
`(CVD). Next the wafer is masked with a polymer photoresist (PR), a template
`with the pattern to be etched onto the SiO2 layer is placed over the PR and
`the wafer is exposed to ultraviolet irradiation. If the mask is a positive PR,
`the light will cause scission in the polymer so that the exposed areas will
`dissolve when the wafer is placed in the developer. On the other hand, when
`a negative PR mask is exposed to ultraviolet irradiation, crosslinking of the
`polymer chains occurs and the unexposed areas dissolve in the developer.
`The undeveloped portion of the PR (in either case) will protect the covered
`areas from etching.
`After the exposed areas of SiO; are etched to form trenches (either by
`wet etching (see P5—ll) or plasma etching), the remaining PR is removed.
`Next the wafer is placed in a furnace containing gas molecules ofthe desired
`LAM Exh 1008-pg 3_
`
`LAM Exh 1008-pg 3
`
`

`
`Diffusion and Reaction in Porous Catalysts
`
`C*a:
`
`T Pulling Mechanism
`seed Crystal
`
`Polishing
`Slurry
`
`Silicon Melt in
`8 uartz Crucibl =
`
`Czochralski Crystal Growth
`
`Wafer Slicing
`
`Wafer Cleaning and Polishing
`
`Photoresist
`
`SWCSOY; Di°Xlde
`Photoresist Application
`1
`UV Irradiation
`
`l
`Photoresist
`Silicon Dioxide
`
`Silicon Dioxide
`"‘J
`CVD of Silicon Dioxide
`
`Clean, Polished Silicon Wafa
`
`Mask
`
`C 2 !
`Silicon Dioxide
`
`____ I 2 2
`
`Photoresist Exposure
`
`Photoresist Development.
`
`Etch and then Remove Phcj
`
`l
`‘-:_-:'_-_':I‘-
`
`CVD (two films)
`
`Silicon Dioxide Etching
`
`Doping by Phosphorus D5:
`
`Mask, Etch, then Strip Mask
`
`CVD, Mask, Etch, Strip Mask
`
`CVD of Final L331:
`
`Figure 11-11 Microelectronic fabrication steps.
`
`dopant, which then diffuse into the exposed silicon. After diffusion of
`to the desired depth in the wafer it is removed and then covered with
`by CVD. The sequence of masking, etching, CVD, and metallizatim
`tinues until the desired device is formed. A schematic of a final chip is -
`in the lower right-hand corner of Figure 11-1 1.
`One of the key steps in the chip making process is the de ‘ '
`different semiconductors and metals on the surface of the chip. ‘IT:
`can be achieved by CVD. CVD mechanisms were discussed in C
`consequently this section will focus on CVD reactors. A number (I ~
`reactor types have been used, such as barrel reactors. boat TCHCIL
`
`LAM Exh 1008-pg 4
`
`

`
`Sec. 11.8
`
`Chemical Vapor Deposition (CVD) Reactors
`
`Support Boat
`
`Figure 11-12 LPCVD boat reactor.
`
`horizontal and vertical reactors. A description of these reactors and modeling
`equations are given by Jensen.6
`One of the more common CVD reactors is the Horizontal Low Pressure
`
`CVD (LPCVD) reactor. This reactor operates at pressures of approximately
`100 Pa. The main advantage of the LPCVD is its capability of processing a
`large number of wafers without detrimental effects to film uniformity. Owing
`to the large increases in the diffusion coefficient at low pressures (recall
`Table 10-2) surface reactions are more likely to be controlling than mass
`transfer. A schematic of a LPCVD reactor is shown in Figure 11-12.
`To illustrate LPCVD modeling we shall use a specific but simplified
`example, the deposition of silicon from a gas stream of SiI-I2. The reaction
`mechanism is
`
`CVD Reaction
`Sequence in Silicon
`Deposition
`
`SiH2(g) + S <:’ SiH2'S
`_
`_
`SlH2'S *’ Sl(S) + H2'S
`
`H2-S 1 S + H2(g)
`
`Here we have assumed that the equilibrium for the dissociation of SiH4
`discussed in Problem 6-3 lies far to the right.
`The corresponding rate law is
`
`klPSiH1
`"; = ——e'
`1 + KIPH2 + K2PSiH2
`'5
`
`(11-75)
`
`Recalling that the adsorption constants K , and K2 decrease with increasing
`temperatures, an excellent approximation at high temperature is
`
`1 >(K1PH; + K2PSiH:)
`
`consequently, the deposition rate can be modeled as first-order in SiH2, i.e.,
`
`where A E SiH2
`
`T
`
`"st 5 k1PSiH2 2 kCSiH2 E /(CA
`
`(11‘76)
`
`6 K. F. Jensen. Chemical Engineering Science. 42. 923 (1987).
`
`L 5 I
`
`5
`
`LAM Exh 1008-pg 5
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`

`
`Diffusion and Reaction in Porous Catalysts
`
`37::
`
`PRESSURE
`SENSOR
`
`3-ZONE
`
`TEMPERRTURE
`CONTROL
`
`TRRP E?
`UFICUUM
`
`GRS CONTROL
`SYSTEM
`
`Figure 11-13 LPCVD boat reactor with peripherals.
`
`‘-.-
`Modeling Concepts We shall model the axial flow in the annular
`' _
`being laminar. This assumption is reasonable because a typical
`number for flow in a LPCVD reactor is less than 1. As the react:-.1
`
`flow through the annulus, the reactants diffuse from the annulus w
`inward between the wafers to coat them.7
`
`The reacting gas flows through the annulus between the outer _
`of the cylindrical wafers and the tube wall (see Figure 11-13). The
`sponding cross-sectional area of the annulus is
`
`Ar" = “(R12 _
`
`V
`where R, and R... are the radii of the tube and wafer, respectively.
`SiH2 is being consumed by CVD. the mole fraction of SiH2 (i.e., the —
`in the annulus, yAA, decreases as the reactant flows down the len
`annulus.
`
`Flow in the
`Annulus
`
`7 K. F. Jensen, J. Electroclzemical S0c'iet_v, 130, 1450 (1983).
`
`LAM Exh 1008-pg 6
`
`

`
`The reacting gases diffuse out of the annular region into the space
`between the wafers where the mole fraction is represented by yA. As mol-
`ecules diffuse radially inward some of them are adsorbed and deposited on
`the wafer surface. The reaction products then diffuse radially outward into
`the gas stream axially flowing in the annulus. This system can be analyzed
`in a manner analogous to flow through a packed catalyst bed where the
`reaction gases diffuse into the catalyst pellets. In this analysis we used an
`effectiveness factor to determine the overall rate of reaction per volume (or
`mass) of reactor bed. We can extend this idea to LPCVD reactors where
`the reactants diffuse from the annular flow channel radially inward between
`the wafers.
`
`11.8.1 Effectiveness Factor for a LPCVD Reactor
`
`Silicon will deposit on the wafers, the reactor walls, and on the boat
`support. Deposition on the walls and support will take place at the reactant
`concentrations in the annulus. However. the concentration of A between
`the wafers is less than the concentration in the annulus. Consequently, the
`rate of deposition on the wafer will be less than the rate at conditions in the
`annulus. Fortunately these two concentrations can be related by the effec-
`tiveness factor. We can determine the effectiveness factor once the con-
`
`centration profile in the region between the wafers is obtained.
`
`U
`
`Actual rate of reaction
`
`_ Rate of reaction when entire wafer surface is exposed to
`the concentration in the annulus CAA (i.e., yAA)
`
`(1 1-77)
`
`R“.
`
`= 2]‘-)
`
`W
`
`. _ ."
`
`2m(
`
`.
`
`.
`
`’A(,)) all : 2"”Ru-€(“WAr
`
`iI‘=R..»)
`
`dCA
`
`AB dr
`
`I'=Rn
`
`21rR?.-(-r’.StA)
`
`21TR?.»( — I'M.)
`
`R...( - r’/RA)
`
`(1 1-78)
`
`where 6 is the distance between wafers. and —rL§A is the rate of disappear-
`ance of A at the concentration of A in the annular region. CAA. We now
`use 1] to express the actual rate of reaction per unit surface area of wafer
`in terms of the rate of reaction at conditions in the annulus.
`
`Actual Rate = —r’,i\An
`
`(11-79)
`
`Letting ‘a’ be the wafer surface area per unit volume of reactor, the
`rate of consumption of species A by the wafer per unit volume of reactor
`is
`
`—rA... = —I'1&AcIn
`
`(11-80)
`
`LAM Exh 1008-pg 7
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`LAM Exh 1008-pg 7
`
`

`
`640
`
`Diffusion and Reaction in Porous Catalysts
`
`Che:
`
`Example 11-4 Diffusion Between Wafers
`
`Derive an equation for the reactant gas concentration as a function of wafer
`radius and then determine the effectiveness factor.
`
`SiH2(g) ——> Si(S) + H2(g)
`
`In terms of the diffusing gas phase components, we can write this reaction as:
`
`A(g) ~—> 13(8)
`
`Solution
`
`W
`
`A’ r+Ar
`
`Diffusion between
`
`the wafers
`
`I
`
`wA
`
`Figure Ell-4.1
`
`The shell balance on the reactant diffusing between two wafers separated b}
`a distance 6 shown in Figure E11-4.1 gives
`
`In
`
`—
`
`Out
`
`+ Generation = 0
`
`W,.,,.2m-6 |, — W,\,.2'rrr€
`
`[,.+_\,. + 2(21'rrAI‘ I'LL.-) = 0
`
`where r',g.,. = rate of generation of species A per unit wafer surface area. The
`factor of 2 appears in the generation term because there are two wafer surfaces
`exposed in each differential volume element. Dividing by 2m-Aré. taking the
`limit as Ar approaches zero, and then rearranging gives
`
`1 d(WA,-I‘)
`-
`r
`dr
`
`=
`
`2I’A..-
`6
`
`El1--1.!-
`
`(
`
`Recalling the constitutive equation for the molar flux WA,‘ in radial coordinates
`
`[ 1
`
`Wm = _CDAB
`
`+ ,vA<WA,. + WB:-)
`
`(E11-4.:
`
`For every one molecule of SlHg (i.e., species A) that diffuses IN one molecuk
`of H3 (i.e., species B) diffuses OUT.
`
`WEI‘
`
`— WAr
`
`WA, = —cDAB
`
`—
`
`_
`
`(El1-4.E-
`
`LAM Exh 1008-pg 8
`
`

`
`Diffusion with
`
`reaction between
`
`wafers
`
`For a first-order reaction
`
`Substituting equations (11-76) and (E11—4.3) into Equation (E11-4.1) we get
`
`—r’,§,,. = kCA
`
`(11-76)
`
`1
`
`av
`
`— —_ <
`
`r dCA
`
`2/\. CA
`
`(E11-4.4)
`
`" d’
`
`The corresponding boundary conditions are
`
`At I’ = R“.
`
`CA = CAA
`
`At r
`
`0
`
`Ea
`dr
`
`— 0 and CA is finite
`
`(E11-4.6)
`
`Let )\ = F/R“. and ‘I’, = CA/CAA
`
`then
`
`(E11-4.7)
`
`where cl)? = l2)kR‘-2AB
`
`The boundary conditions are
`
`at A
`
`0:
`
`513
`d)\
`
`— 0 and ‘If is finite
`
`at>\=l
`
`‘ll =l.0
`
`Equation (E11-4.7) is a form of BesseI‘s Equation.
`
`The general form of the solution to Bessel’s equation is8
`
`‘I’ = AL.(<bi>\) + BKa(d>i>\)
`
`(E11-4.8)
`
`Where la is a modified Bessel function of the first kind of order zero and
`K0 is a modified Bessel function of the second kind of order zero. The
`second boundary condition requires ‘I’ to be finite at A = 0. Therefore
`B must be zero because K,,(0) = 00. Using the first boundary condition
`we get 1 = AIa(d).), then A = 1/I(,(d>,). The concentration profile in the
`space between the wafers is
`
`CA =
`‘I’ = CAA
`
`I..<d>.)
`
`7, =
`
`,-: “(Z R..«€
`-W ,-
`21TRI\‘(—’AA
`
`(E11—4.10)
`
`8 Mickley, H. S., T. K. Sherwood. and C. E. Reed. Applied Matlzenzatics in Chem-
`ical Engineering, p 174, McGraw-Hill. New York. (1957).
`
`LAM Exh 1008-p
`
`LAM Exh 1008-pg 9
`
`

`
`Diffusion and Reaction in Porous Catalysts
`
`SW3:
`
`— < - DAB ‘E?
`V'=Ru
`,
`n=—j———————=2
`2/\’C‘AATl’R,’,.
`
`)<2wR...e)
`
`1“:
`dk A :.
`
`2kR?..
`owe
`
`E11~:.::.
`
`(
`
`‘/_‘1’
`dx /\:l
`§
`(bi
`
`:7
`
`Tl
`
`Tl
`
`21i(<bi)
`: d)l[o(¢1)
`
`(EH49
`
`The concentration profile along the radius of the wafer disk as uel I
`the wafer shape is shown in Figure El 1-4.2 for different values of the T31
`modulus.
`
`Low :1)
`
`Medium :1»
`
`Th':'
`
`L3‘
`
`Meouim
`
`II
`
`Hill’!
`
`Figure E11-4.2 Radial concentration profile.
`
`Accounting for Si
`deposition on the
`walls and support
`
`Deposition on the Peripherals Silicon will deposit on the walls of a e-
`and on the boat support in addition to the wafers. This rate of d _
`on the walls and support is
`
`"'§i.,, = ‘FAA = (1 ‘l’ a)kCAA
`
`(1:
`
`surface area of boat support
`surface area of tube
`
`Owing to high temperature and low pressure. radiation is the do ‘ ‘
`-
`transfer mechanism, therefore small temperature differences exist
`the wafer and reactor wall. Consequently, there is no need to C; l
`mole and energy balances for these small temperature gradients.
`
`LAM Exh 1008-pg 10