Growth of Compound Semiconductors and Superlattices by

Chapter 19 Growth of Compound Semiconductors and Superlattices by Organometallic Chemical Vapor Deposition Transport Phenomena

Downloaded by CORNELL UNIV on September 29, 2016 | http://pubs.acs.org Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch019

Klavs F. Jensen, Dimitrios I. Fotiadis, Donald R. McKenna, and

Harry K. Moffat Minnesota Supercomputer Institute and Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 Large scale computations are used to simulate the growth of GaAs and AlGaAs thin films by organometallic chemical vapor deposition (MOCVD) in horizontal and vertical reactors. The computations provide newinsightsintoflow,heat and mass transfer effects on film thickness uniformity and interface abruptness in superlattice growth. For the horizontal reactor, the simulations demonstrate the existence of longitudinal buoyancy drivenflowrolls that adversely affect film thickness uniformity. For the vertical reactor, the film thickness uniformity is shown to be influenced by susceptor edge, susceptor rotation, reactor wall and buoyancy effects. It is demonstrated that nonlinear interactions between the transport processes lead to the existence of multiple steady flowsincertain operating regions. Concentration transients in the growth of AlAs/GaAs interfaces are simulated and it is shown that the presence offlowrecirculation cells widens the interface. Processing of electronic materials ls essentially the fabrication of artificially microstructured materials with unique electronic and optical properties. This ls accomplished through successive film deposition, pat>ternlng, etching and doping operations. Depending on the complexity of the final Integrated circuit, the manufacture of the final microstructure may take up to several hundred process steps each of which Involves complex physical transport processes and chemical reactions In threedimensional domains. As the level of circuit Integration Increases, the minimum device feature size shrinks, and material constraints escalate, It ls becoming Increasingly costly and difficult to modify processes by design rules and one-parameter-at>-a-time-experiments. For example, the fabrication of state-of-the-art multiple quantum well devices for ultrafast electronic and optical components requires the growth of a 5 nm thin film of a 0097-6156/87/0353-0353S06.75/0 © 1987 American Chemical Society Jensen and Truhlar; Supercomputer Research in Chemistry and Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1987.


354

SUPERCOMPUTER RESEARCH

c o m p o u n d s e m i c o n d u c t o r u n i f o r m l y o v e r a 50 m m substrate. T h u s , there ls a s t r o n g Incentive to use detailed m a t h e m a t i c a l m o d e l s to predict process p e r f o r m a n c e . In o r d e r for these m o d e l s to be useful In process design, o p e r a t i o n a n d c o n t r o l , t h e y m u s t give an accurate picture o f the u n d e r l y i n g time d e p e n d e n t h y d r o d y n a m i c s , energy transfer, mass transport a n d c h e m i c a l reactions w i t h i n the processing systems. T h i s m e a n s m o v i n g away f r o m o v e r s i m p l i f i e d analytical m o d e l s towards detailed m o d e l s In the f o r m of m u l t i p l e n o n l i n e a r , c o u p l e d partial differential equations In time a n d three spatial d i m e n s i o n s . T h i s necessarily entails large scale c o m p u t a t i o n s s i m i lar to those e n c o u n t e r e d In c o m b u s t i o n a n d atmospheric c h e m i s t r y m o d e l ling. In this paper we present s u p e r c o m p u t e r s i m u l a t i o n s o f transport p h e n o m e n a In the g r o w t h o f c o m p o u n d s e m i c o n d u c t o r s , In particular GaAs and A l G a A s by organometallic chemical vapor deposition ( M O C V D ) . T h i s technique has r e c e i v e d considerable attention because o f Its potential f o r large scale p r o d u c t i o n o f o p t i c a l a n d digital device structures ( 1 - 3 ) . A s the name M O C V D Indicates, the process entails the gas phase transport o f o r g a n o m e t a l l i c precursors o f at least one o f the film constituents to a heated substrate where the film ls f o r m e d . C o m m o n o v e r a l l reactions In M O C V D are: Ga(CH ) 3

xAl(CH ) 3

3

3

4- A s H

3

GaAs + 3 C H

+ (l-x)Ga(CH3)3+ A s H

3

(la)

4

-+ A ^ G a ^ A s + 3 C H

4

(lb)

F i g u r e 1 Illustrates two general M O C V D reactor configurations, the hori z o n t a l reactor and the a x l s y m m e t r i c v e r t i c a l reactor. T h e reactant gas ( A s H , G a ( C H ) and A 1 ( C H ) ) enters c o l d and heats up as It flows t o w a r d the substrate where a s o l i d film o f A l G a A s ls b e i n g deposited. T h e c h e m i c a l transformations I n v o l v e d In the d e p o s i t i o n process m a y o c c u r b o t h In the gas phase a n d o n the surface o f the g r o w i n g film. T h e usefulness o f a particular film o r heterostucture (I.e. an assembly o f A l m s o f different c o m p o s i t i o n s ) depends o n several factors I n c l u d i n g : (1) the l e v e l a n d spatial d i s t r i b u t i o n o f dopants ( I n t e n t i o n a l l y as w e l l as u n i n t e n t i o n a l l y added), (11) film thickness u n i f o r m i t y o v e r the substrate, a n d (111) the Interface abruptness between layers o f different c o m p o s i t i o n s (3.). These performance criteria are s t r o n g l y affected by the flow profiles w i t h i n the reactor e n c l o s u r e . E n t r a n c e effects a n d b u o y a n c y d r i v e n flows, caused b y large t h e r m a l differences between w a l l a n d susceptor temperatures, are k n o w n to produce c o m p l e x mass transfer patterns leading to severe g r o w t h rate a n d film c o m p o s i t i o n n o n u n l f o r m l t l e s ( 4 . 5 ) . C o n s e q u e n t l y , there has been considerable Interest In e x p e r i m e n t a l o b s e r v a t i o n s o f flow structures In M O C V D reactors. F l o w v i s u a l i z a t i o n s w i t h T 1 0 s m o k e have r e v e a l e d the existence o f l o n g i t u d i n a l rolls a n d a particle free r e g i o n adjacent to the surface In h o r i z o n t a l reactors ( 6 - 9 ) . T h i s r e g i o n r e s u l t i n g f r o m t h e r m o p h o r e t l c transport o f T 1 0 particles away f r o m the h o t substrate has often been m i s i n t e r p r e t e d as a stagnant layer (lu). T10 s m o k e e x p e r i m e n t s have also d e m o n s t r a t e d the existence o f r e c i r c u l a t i o n 3

3

3

3

3

2

2

2

Jensen and Truhlar; Supercomputer Research in Chemistry and Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1987.


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355

Inlet

(a)

Figure

1.

^ (b)

T w o typical M O C V D reactor configurations; (a) i z o n t a l reactor (b) vertical reactor.


hor-



356

cells In v e r t i c a l reactors (11*12.). L a s e r h o l o g r a p h y o f a x l a l l y averaged d e n s i t y gradients In h o r i z o n t a l reactors a n d radially average density gra dients In v e r t i c a l reactors have further d e m o n s t r a t e d that the M O C V D reactor flows m a y be s t r o n g l y p e r t u r b e d b y entrance a n d b u o y a n c y effects (4*13). The e x p e r i m e n t s have l e d to considerable Insights Into M O C V D reac t o r flows, b u t a complete u n d e r s t a n d i n g o f the detailed flow structure ls s t i l l m i s s i n g and It ls n o t l i k e l y to be a c h i e v e d t h r o u g h e x p e r i m e n t s alone. S m o k e testing suffers f r o m artifacts I n t r o d u c e d by t h e r m o p h o r e tic tran s p o r t o f the s e e d particles w h i l e Interference h o l o g r a p h y gives spatially averaged d e n s i t y gradients. T h u s , transport p h e n o m e n a m o d e l l i n g ls a necessary c o m p o n e n t In u n d e r s t a n d i n g a n d d e s i g n i n g M O C V D reactors capable o f p r o d u c i n g large area u n i f o r m i t y a n d sharp Interfaces. I n the f o l l o w i n g sections we s u m m a r i z e recent large scale finite e l e m e n t c o m p u tations o f detailed t w o - a n d t h r e e - d i m e n s i o n a l m o d e l s for the two classical M O C V D reactor configurations Illustrated In F i g u r e 1. Modelling Equations M O C V D reactor m o d e l s consist o f m o m e n t u m , energy and species balances w i t h c o r r e s p o n d i n g b o u n d a r y c o n d i t i o n s . T h e general f o r m o f these balance equations are g i v e n In standard transport p h e n o m e n a texts (1±)Since the reactants In M O C V D are u s e d dilute In H o r s o m e Inert carrier gas (e.g. H e ) , v o l u m e e x p a n s i o n due to the change In n u m b e r o f m o l e s b e t w e e n reactants a n d products ls negligible. I n a d d i t i o n , energy c o n t r i b u t i o n s caused b y the heat o f reaction are Insignificant. T h e r e f o r e , It ls reasonable to separate the flow a n d heat transfer p r o b l e m f r o m the r e a c t l o n - c o n v e c t l o n - d l f f u s l o n p r o b l e m w h i c h leads to r e d u c e d storage a n d C P U t i m e r e q u i r e m e n t s . It also has the a d d e d advantage that different c h e m i c a l m e c h a n i s m s a n d k i n e t i c s can be c o n s i d e r e d f o r the same flow s i m u l a t i o n . E v e n If the p r o b l e m s were w e a k l y c o u p l e d , a P i c a r d type Iteration m i g h t be c o m p u t a t i o n a l l y m o r e efficient t h a n s o l v i n g the fully c o u p l e d equations together. E x p a n s i o n effects due to d e n s i t y changes In the gas phase play a major role In the flow b e h a v i o r a n d have to be I n c l u d e d In the c o m p u t a t i o n s . H o w e v e r , because o f the l o w M a c h n u m b e r s o f M O C V D flows, It ls reasonable to neglect c o m p r e s s i b i l i t y effects w h i c h otherwise w o u l d u n n e c e s s a r i l y complicate the c o m p u t a t i o n s . Since the film g r o w t h rate In c h e m i c a l v a p o r d e p o s i t i o n ls slow c o m p a r e d to the gas phase d y n a m i c s , the flow d y n a m i c s m a y be a s s u m e d to be In pseudo-steady state, I.e. the t i m e d e r i v a t i v e s In the m o m e n t u m a n d e n e r g y balance m a y be set to z e r o . T h e flow d e s c r i p t i o n t h e n takes the form: 2

> ( v v v ) = S7'T + PZ » τ = /ι v v + ( V v )

T

-

2 —μ V * v + Ρ I 3

(2)

w h e r e P , p, a n d μ are the pressure, density a n d v i s c o s i t y , respectively, g represents the acceleration due to gravity a n d ν ls a v e c t o r o f the v e l o c i t y


19.

JENSEN ET AL.

Compound

Semiconductors

components. The m o m e n t u m o v e r a l l c o n t i n u i t y balance:

balance

must

and Superlattices be

357

combined with

V'pv=0

the

(3)

a n d the Ideal gas law as an e q u a t i o n o f state: ρ =P M /RT 0

(4)

W

Since v a r i a t i o n s In the pressure Induced by fluid d y n a m i c effects are negli gible for M O C V D reactor flows, the Inlet pressure, P , ls u s e d . I n f o r m u lating the energy balance, the c o n t r i b u t i o n s f r o m pressure changes, v i s c o u s dissipation a n d D u f o u r effects m a y neglected for M O C V D c o n d i tions (14.15) so the e q u a t i o n b e c o m e s :


0

pC-pVVT = v ( k v T )

+ Σ Hi

V'Ji

(5) j=i

i = i

w h e r e C , k, a n d Τ are the heat capacity, t h e r m a l c o n d u c t i v i t y a n d t e m perature, respectively. T h e s e c o n d t e r m o n the right-hand-side represents the c o n t r i b u t i o n f r o m gas phase reactions. T h i s can be neglected since M O C V D ls u s u a l l y p e r f o r m e d w i t h a large excess o f carrier gas a n d the enthalpies o f reactions, H j , are s m a l l c o m p a r e d to c o m b u s t i o n reactions. In a d d i t i o n , s i m p l e F l c k l a n diffusion rather t h a n m u l t i c o m p o n e n t diffusion m a y be u s e d to f o r m u l a t e balances o v e r the dilute reactants: p

#Cj — + cv-vxj = V c D

r i

m

|yxi + k

η n V In T J + £ J i/jfRf e

T

(6)

g

where J = l , . . . , n ls the n u m b e r o f gas phase reactions a n d 1=1,...,S-1 n u m b e r species. Since the m o l e fractions m u s t s u m to u n i t y , o n l y S - l equations n e e d to be s o l v e d , c ls the total c o n c e n t r a t i o n a n d Xj ls the m o l e fraction o f c o m p o n e n t 1. T h e s u m o n the rlght-hand-slde represents the p r o d u c t i o n o f species 1 In n gas phase reactions. D a n d k are the F l c k l a n diffusion coefficient a n d the t h e r m n a l diffusion ratio, respectively. Because o f the often large t h e r m a l gradients In C V D , the S o r e t effect ( t h e r m a l diffusion) m a y contribute to the o v e r a l l g r o w t h b e h a v i o r ( 1 6 . 1 7 ) . b u t It w i l l n o t be c o n s i d e r e d f u r t h e r In this s u m m a r y paper. g

i

m

T

N o slip ls u s e d as the v e l o c i t y b o u n d a r y c o n d i t i o n s at all w a l l s . A c t u ally there ls a finite n o r m a l v e l o c i t y at the d e p o s i t i o n surface, b u t It ls Insignificant In the case o f dilute reactants. T h e Inlet flow ls a s s u m e d to be P o l s e u l l l e flow w h i l e zero stresses are specified at the reactor e x i t . The b o u n d a r y c o n d i t i o n s for the temperature play a central role In C V D reac t o r b e h a v i o r . H e r e we e m p l o y Idealized b o u n d a r y c o n d i t i o n s In the absence o f detailed heat transfer m o d e l l i n g o f an actual reactor. T w o wall c o n d i t i o n s w i l l be c o n s i d e r e d : (1) adlabatlc side walls, I.e. d T / d n = 0, and (11) fixed side w a l l temperatures c o r r e s p o n d i n g to c o o l e d reactor walls. F o r the reactive species, no net n o r m a l flux ls specified o n n o n r e a c t i n g surfaces. A t substrate surface, the flux o f the T t h species equals the rate o f reaction o f 1 In n surface reactions, I.e. s



358

V X j + k i - V In Τ

T o complete the m o d e l l i n g equations, k i n e t i c rate expressions a n d tran s p o r t coefficients m u s t be specified. H o w e v e r , the e l e m e n t a r y kinetics of the M O C V D reactions [ l a and l b ] are largely u n k n o w n . N e v e r t h e l e s s , since e x p e r i m e n t a l o b s e r v a t i o n s have s h o w n that M O C V D o f G a A s at a t m o s p h e r i c c o n d i t i o n s ls c o n t r o l l e d by mass transfer o f the group III c o n t a i n i n g species (IS.), It ls possible to describe the g r o w t h rate b y a group III species ( G a ( C H ) a n d / o r A 1 ( C H ) ) diffusion m o d e l w i t h fast surface r e a c t i o n . T h e o b s e r v a t i o n s leading to this c o n c l u s i o n Include the Indepen dence o f g r o w t h rate o n the partial pressure o f A s H , the first o r d e r dependence o n the partial pressure o f G a ( C H ) , a n d the s m a l l depen dence o f g r o w t h rate o n substrate temperature (2JL&). 3


(7)

·η

3

3

3

3

3

3

Numerical Solution The c o m b i n e d fluid flow, heat transfer, mass transfer a n d reaction p r o b l e m , d e s c r i b e d b y E q u a t i o n s 2-7, lead to t h r e e - d i m e n s i o n a l , n o n linear, t i m e d e p e n d e n t partial differential equations. T h e general n u m e r i cal s o l u t i o n o f these goes b e y o n d the m e m o r y a n d speed capabilities o f the c u r r e n t generation o f s u p e r c o m p u t e r s . T h e r e f o r e , we c o n s i d e r appropriate physical assumptions to reduce the c o m p u t a t i o n s . F o r typical flow rates In the h o r i z o n t a l reactor ( F i g u r e l a ) It ls reason able to assume that the l o n g i t u d i n a l diffusion o f m o m e n t u m ls s m a l l In c o m p a r i s o n w i t h the a d v e c t l o n . T h i s Implies that the steady state flow a n d temperature profiles can be c o m p u t e d b y m a r c h i n g a t w o - d i m e n s i o n a l ( 2 D ) finite e l e m e n t discretization o f the transverse reactor s e c t i o n along the l e n g t h o f the reactor. T h u s , the o r i g i n a l steady state threed i m e n s i o n a l ( 3 D ) p r o b l e m has been t r a n s f o r m e d Into a f o r m that ls e q u i v a l e n t to a 2 D t i m e d e p e n d e n t transport p r o b l e m . T h e G a l e r k l n F i n ite e l e m e n t m e t h o d (UL) was e m p l o y e d to dlscretlze the equations In the transverse d i r e c t i o n . I n the flow p r o b l e m , E q u a t i o n s 2-5, m i x e d o r d e r Interpolation was u s e d ( l f L 2 ! l ) . T h e r e s u l t i n g set o f n o n l i n e a r o r d i n a r y differential equations a n d algebraic constraints were Integrated along the l e n g t h o f the reactor b y the differential-algebraic e q u a t i o n s o l v e r , D A S S L ( 2 1 ) . I n the r e a c t i o n - a d v e c t l o n - d l f f u s l o n p r o b l e m b i l i n e a r elements were u s e d . A different m e s h a n d Integration steps t h a n the flow p r o b l e m were e m p l o y e d to i m p r o v e the efficiency o f the s o l u t i o n . T h e transverse v e l o cities a n d temperatures n e e d e d In the mass transfer code were c o m p u t e d by Interpolation In the finite e l e m e n t s o l u t i o n w h i l e Interpolation In the axial d i r e c t i o n o f the v e l o c i t y a n d temperature field b e t w e e n flow p r o g r a m Integration steps was done w i t h the same p o l y n o m i a l u s e d b y D A S S L to a p p r o x i m a t e the flow s o l u t i o n . T h e flow a n d mass transfer p r o b l e m s each typically I n v o l v e d 3000 u n k n o w n s a n d t o o k 5-10 m i n u t e s o f C r a y - 2 C P U time. The v e r t i c a l reactor ( F i g u r e l b ) ls a s s u m e d to be a x l s y m m e t r l c w h i c h



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leads to a 2 D f o r m u l a t i o n . T h e p r o b l e m r e m a i n s 2 D w h e n susceptor rota t i o n ls I n c l u d e d , b u t three m o m e n t u m balances c o r r e s p o n d i n g to the radial, a z l m u t h a l a n d axial directions have to be s o l v e d ( 2 2 - 2 4 ) . The G a l e r k l n finite e l e m e n t m e t h o d was u s e d to dlscretlze the equations In the same m a n n e r as done for the transverse plane In the h o r i z o n t a l reactor ( 1 6 . 2 2 . 2 4 ) . I n o r d e r to s i m u l a t e m o r e realistic reactor shapes t h a n a s i m ple rectangular d o m a i n , the weak constraint m e t h o d o f R y s k l n a n d L e a l ( 2 £ ) was u s e d to generate o r t h o g o n a l grids for the r e a c t i o n d o m a i n s . F i g ure 2 s h o w s e x a m p l e s o f grids u s e d for reactor shapes that w i l l be c o n s i d e r e d In the results s e c t i o n . T h e G a l e r k l n finite e l e m e n t d i s c r e t i z a t i o n leads to a set o f n o n l i n e a r algebraic equations o f the general f o r m : G(u,p)

=0

(8)

w h e r e u ls the v e c t o r o f the v a l u e s o f the d e p e n d e n t variables at the node points and ρ ls the v e c t o r o f parameters. T h i s set o f equations can be s o l v e d b y N e w t o n - R a p h s o n Iteration, I.e. Gu(«

( n )

,p)[u

( n + 1 )

n

-

u( >]=-

n

G(u< \p)

(9)

where dGj Gu ij

^ u "

H o w e v e r , because o f the s t r o n g n o n l l n e a r l t l e s In the reactor flow prob l e m , c o n t i n u a t i o n procedures m u s t be u s e d to o b t a i n a g o o d Initial guess for the N e w t o n Iteration. A s i m p l e first o r d e r c o n t i n u a t i o n s c h e m e falls at flow t r a n s i t i o n points (bifurcations) where the J a c o b i a n , G ^ , b e c o m e s s i n g u l a r . T o c i r c u m v e n t this p r o b l e m an arclength c o n t i n u a t i o n scheme d i s c u s s e d b y K e l l e r (26.27) a n d C h a n (2&) ls u s e d w h i c h leads to the Inflated s y s t e m : u

(n+l) _

u

(n)

Gin)'

λ

(η+1)

λ

(η)

N

n

G< > N

N/, >

N

in)

-

(n)

(10)

w h e r e λ ls the c o n t i n u a t i o n parameter, s ls the arclength parameter a n d Ν ls a l i n e a r i z a t i o n o f the arclength c o n d i t i o n , I.e. T

N(Ù(S),X(S),S) = Û ( S ) ( U ( S ) 0

u(s )) 0

+ Ms )(\(s) - X ( s ) ) - ( s - s ) = 0 0

0

0

(11)

T h e Inflated system (10) ls n o n s l n g u l a r e v e n w h e n the J a c o b i a n o f the finite e l e m e n t f o r m u l a t i o n (8) b e c o m e s s i n g u l a r (2Û.) at s i m p l e t u r n i n g points. Since the s o l u t i o n changes stability at the t u r n i n g points, these are I m p o r t a n t to the u n d e r s t a n d i n g o f the o v e r a l l reactor b e h a v i o r . I n p r i n c i ple, the Inflated system m a y be u s e d to d e t e r m i n e the b i f u r c a t i o n points a n d s w i t c h s o l u t i o n branches b y Increasing the arclength parameter, s, and




360

F i g u r e 2.

C o m p u t e d grids for f o u r different shapes o f a v e r t i c a l , a x l s y m m e t r l c M O C V D reactor.


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Compound

Semiconductors

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Superlattices

361

r e f i n i n g the step size w h e n the J a c o b i a n , G , b e c o m e s s i n g u l a r (I.e. det = 0 ) . H o w e v e r , this o f t e n leads to e x c e s s i v e l y m a n y c o m p u t a t i o n s . T h e r e f o r e , we use a quadratlcally c o n v e r g e n t m e t h o d based o n applying N e w t o n ' s m e t h o d to the characterization, d X ( s ) / d s = 0 (2S.). T h i s procedure w o r k s w e l l f o r locating s i m p l e t u r n i n g points present In the m i x e d c o n v e c t i o n flow g o v e r n i n g v e r t i c a l reactors. T h e f o l l o w i n g e q u a t i o n gives an a p p r o x i m a t i o n to the s e c o n d d e r i v a t i v e w.r.t. arclength o f the bifurcat i o n p a r a m e t e r near the t u r n i n g p o i n t : u

< G (s )u(s )u(s )

Gu(so) G x ( s ) u(s ) N ( s ) N ( s ) [Ms ) 0

0


u

0

x

0

uu

=

~

0

0

0

0

]N u u ( s 0 ) ù ( s 0 ) û ( s 0 )

+ 2G (s )ù(s )X(s ) + Gxx(s )X(s ) u X

0

0

0

0

+ 2 N ( s ) ù ( s ) M s ) + Nxx(s )X(s ) u X

0

0

0

0

2

0

2

0

+ N (s ) s s

(12)

0

F o l l o w i n g C h a n (2S.) a difference a p p r o x i m a t i o n ls u s e d to c o m p u t e the s e c o n d d e r i v a t i v e s o f G . A N e w t o n Iteration ls t h e n a p p l i e d to the equation 0 = X(s ) 0

n

n

=X(s ) + X(s )(s

n + 1

-

n

s )

(13)

to d e t e r m i n e the t u r n i n g p o i n t , I.e. where X ( s ) = 0 . 0

T h e v e r t i c a l reactor s i m u l a t i o n s r e p o r t e d In this paper typically I n v o l v e d 14,000 u n k n o w n s a n d t o o k 25 C P U seconds per N e w t o n Iterat i o n o n a C r a y - 2 . T h e tracing o f a complete f a m i l y o f s o l u t i o n s f o r one p a r a m e t e r (e.g. susceptor t e m p e r a t u r e ) cost a p p r o x i m a t e l y 25 C P U m i n u t e s . T h e latter n u m b e r underscores the advantage o f u s i n g superc o m p u t e r s to u n d e r s t a n d the structure o f the s o l u t i o n space f o r physical p r o b l e m s w h i c h often I n v o l v e m a n y parameters. In the f o l l o w i n g sections we describe s i m u l a t i o n results p r o v i d i n g new Insights Into general M O C V D reactor b e h a v i o r as w e l l as the t w o major practical c o n s i d e r a t i o n s , film u n i f o r m i t y and Interface w i d t h . MOCVD

In H o r i z o n t a l R e a c t o r s

B o u n d a r y layer s i m i l a r i t y s o l u t i o n t r e a t m e n t s have b e e n u s e d extens i v e l y to d e v e l o p analytical m o d e l s for C V D processes (22.). These have b e e n useful In c o r r e l a t i n g e x p e r i m e n t a l o b s e r v a t i o n s (e.g. £.). H o w e v e r , because o f the o v e r s i m p l i f i e d flow d e s c r i p t i o n t h e y c a n n o t be u s e d to extrapolate to new process c o n d i t i o n s o r f o r reactor design. M o r e o v e r , t h e y c a n n o t predict transverse v a r i a t i o n s In film t h i c k n e s s w h i c h m a y o c c u r e v e n In the absence o f s e c o n d a r y flows because o f the presence o f side w a l l s . T w o - d i m e n s i o n a l f u l l y p a r a b o l l z e d transport equations have b e e n u s e d to predict v e l o c i t y , c o n c e n t r a t i o n a n d t e m p e r a t u r e profiles a l o n g the l e n g t h o f h o r i z o n t a l reactors f o r SI C V D ( 1 7 . 3 0 - 3 2 ) . A l t h o u g h these m o d e l s are detailed, t h e y can n e i t h e r capture the effect o f b u o y a n c y d r i v e n s e c o n d a r y flows o r transverse t h i c k n e s s v a r i a t i o n s caused b y the side w a l l s . T h u s , large scale s i m u l a t i o n o f 3 D m o d e l s are n e e d e d to o b t a i n a realistic picture o f h o r i z o n t a l reactor p e r f o r m a n c e .



362

B y u s i n g the 3 D f u l l y parabolic a p p r o x i m a t i o n o f the general transport equations described In the p r e v i o u s s e c t i o n , we have s i m u l a t e d entrance effects a n d the d e v e l o p m e n t o f l o n g i t u d i n a l r o l l patterns In h o r i z o n t a l reactors for g r o w t h o f G a A s (23.) a n d SI (lu). H e r e we r e v i e w the results for two case studies based o n : H as carrier gas, 1 0 0 0 K susceptor, 300 Κ top reactor w a l l temperature, 30 m m reactor height, 60 m m reactor w i d t h , a n d an Inlet v e l o c i t y o f 58 m m / s . T w o Idealized side w a l l t h e r m a l b o u n dary c o n d i t i o n s , (1) no c o o l i n g (adlabatlc) a n d (11) Infinite c o o l i n g , w i l l be u s e d to Illustrate that they s t r o n g l y affect the flow pattern a n d thus the deposition uniformity. Downloaded by CORNELL UNIV on September 29, 2016 | http://pubs.acs.org Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch019

2

T h e case where the side walls are adlabatlc approximates the b e h a v i o r of an air c o o l e d reactor. The entrance flow ls d o m i n a t e d b y the e x p a n s i o n o f gas contacting the h o t susceptor. C o n v e c t i o n rolls, s c h e m a t i c a l l y Illus trated In F i g u r e 3(1), f o r m In the corners a n d e x p a n d along the reactor l e n g t h . T h i s s e c o n d a r y flow ls d r i v e n b y the density gradient between the s l o w e r m o v i n g fluid at the corners a n d the faster a n d therefore less heated fluid In the m l d p l a n e . C o n s e q u e n t l y the rolls rotate towards the mldplane Increasing the mass transfer a n d thus the d e p o s i t i o n rate o f G a A s In the m i d d l e o f the reactor as s h o w n In F i g u r e 3(1). D e p e n d i n g o n the value o f the average R a y l e l g h n u m b e r the rolls w i l l persist w h e n the flow becomes f u l l y d e v e l o p e d . H o w e v e r , e v e n If the rolls are n o t present the side walls affect the transverse film u n i f o r m i t y because the fluid v e l o c i t y decreases to zero at the walls ( 1 6 ) . In the s e c o n d case the side walls are c o o l e d to the top w a l l tempera t u r e . A g a i n the gas e x p a n s i o n d o m i n a t e s the entrance flow a n d the rolls emanate f r o m the corners. Because o f the c o l d side walls the fluid b e c o m e s m o r e dense at the walls t h a n that In the center o f the flow r e g i o n . T h i s produces a r o l l rotating up f r o m the m i d s e c t i o n and d o w n along the walls as Illustrated In F i g u r e 3(11). T h e result ls r e d u c e d mass transfer In the m l d r e g l o n a n d Increased mass transfer near the walls. T h i s ls reflected In the g r o w t h rate s h o w n In F i g u r e 3(11) a n d It ls opposite to the s i t u a t i o n where the side walls are n o t c o o l e d . T h u s , these two e x a m ples demonstrate the k e y role played by the reactor t h e r m a l b o u n d a r y c o n d i t i o n s In o b t a i n i n g film thickness u n i f o r m i t y . These effects have b e e n s h o w n to be In agreement w i t h p u b l i s h e d e x p e r i m e n t a l g r o w t h rate data (23.) and recent, detailed laser D o p p l e r a n e m o m e t r y Investigations s u p p o r t the d e v e l o p m e n t o f l o n g i t u d i n a l c o n v e c t i o n rolls (3±). T r a n s p o r t P h e n o m e n a In V e r t i c a l M O C V D

Reactors

A v e r t i c a l C V D reactor (cf. F i g u r e l b ) consists o f an a x l s y m m e t r l c enclosure w i t h the d e p o s i t i o n surface perpendicular to the I n c o m i n g gas s t r e a m . T h e reactant gases are typically I n t r o d u c e d at the top and flow d o w n towards the heated susceptor. T h u s , the least dense gas ls closest to the g r o w t h Interface w h i c h destabilizes the flow. T h e result ls recircula t i o n cells w h i c h Introduce n o t o n l y film thickness and c o m p o s i t i o n v a r i a tions b u t also b r o a d e n Junctions between layers. T h i s ls particularly of


JENSEN ET AL.

Compound

Semiconductors

and Superlattices

363


19.

Figure

3.

F l o w profiles from G a ( C H ) 3

tor

w i t h (1)

a n d c o r r e s p o n d i n g g r o w t h rate o f 3

and A s H

3

In a h o r i z o n t a l M O C V D

GaAs reac-

no c o o l i n g o f the side walls a n d (11)

cold

(300 K ) side walls; (a) s c h e m a t i c r e p r e s e n t a t i o n o f the longitudinal roll,

(b)

transverse

velocities

50 m m

Into

the reactor, a n d (c) g r o w t h rate o f G a A s (after ref. 33 ).



364


c o n c e r n In g r o w i n g m u l t i p l e q u a n t u m - w e l l structures f o r lasers a n d n o v e l o p t o e l e c t r o n i c devices where one desires abrupt c o m p o s i t i o n changes o v e r a few m o n o l a y e r s . Because o f the n o n u n l f o r m l t y p r o b l e m s caused b y recirculations a n d Inlet flow m a l d i s t r i b u t i o n s , there has b e e n an Interest In r e a l i z i n g Ideal rotating d i s k , I m p i n g i n g Jet o r stagnation p o i n t flows w h i c h have a u n i f o r m mass transfer layer In a r e g i o n a r o u n d the flow stagnation p o i n t (25.). Since the classical analysis o f these flow configurations assumes Infinite substrates and ls o n e - d l m e n s l o n a l , It c a n n o t a c c o u n t f o r w a l l effects, finite susceptor s i z e , a n d b u o y a n c y d r i v e n r e c i r c u l a t i o n s . T h u s , It ls critical to go b e y o n d the classical t r e a t m e n t b y u s i n g the large scale finite e l e m e n t analysis described above In the m o d e l l i n g a n d n u m e r i c a l s o l u t i o n sections. T h e f o l l o w i n g s i m u l a t i o n e x a m p l e s are based o n the s y s t e m parameters: mass transfer c o n t r o l l e d g r o w t h o f G a A s , H carrier gas, 100 T o r r total pressure, 300 Κ Inlet temperature, 900 Κ susceptor, 200 m m Inner d i a m e t e r at the susceptor, 140 m m susceptor d i a m e t e r a n d 120 m m between susceptor a n d Inlet. 2

F i g u r e 4 Illustrates the effect o f v a r y i n g the Inlet flow rate o n flow s t r e a m l i n e s a n d the c o r r e s p o n d i n g radial v a r i a t i o n In the g r o w t h rate rela tive to the g r o w t h rate at the center o f the susceptor. A t the l o w Inlet flow rate ( 7 0 cc/sec (standard c o n d i t i o n s ) ) the flow ls d o m i n a t e d b y a b u o y a n c y d r i v e n r e c i r c u l a t i o n cell above the susceptor, w h i c h reduces the film t h i c k n e s s In the center r e g i o n as s h o w n b y curve ( a ) . A s the Inlet flow rate ls Increased, the b u o y a n c y d r i v e n r e c i r c u l a t i o n c e l l ls e l i m i n a t e d a n d the flow b e c o m e s f o r c e d c o n v e c t i o n d o m i n a t e d a n d resembles an I m p i n g i n g Jet. I n that case the film thickness decreases away f r o m the center o f the susceptor, except f o r a n a r r o w edge r e g i o n . T h u s , It s h o u l d be possible to adjust the Inlet flow v e l o c i t y to give u n i f o r m film t h i c k n e s s . Because o f flow separation a l o n g the reactor w a l l , recirculations can o c c u r e v e n In f o r c e d c o n v e c t i o n situations. These r e c i r c u l a t i o n s have a m i n o r effect o n film t h i c k n e s s u n i f o r m i t y , b u t t h e y c o u l d a d v e r s e l y affect the Interface sharpness In h e t e r o j u n c t l o n g r o w t h . T o this date, m o s t reactor studies have dealt w i t h fixed, regular reac t o r shapes. F o r e x a m p l e , the v e r t i c a l reactor enclosures have been m o d e l l e d as tubes. H o w e v e r , In practice one often tries to m o d i f y the shape o f the enclosure to a v o i d flow separations a n d I m p r o v e u n i f o r m i t y . T h i s procedure ls done b y costly trial a n d error, w h e r e a quartz tube ls made Into a particular shape, cleaned and m o u n t e d In the reactor s y s t e m . S e v e r a l films t y p i c a l l y have to be g r o w n before It ls possible to Judge w h e t h e r o r n o t the c h o s e n shape Is appropriate. T h u s , there ls a consider able practical advantage In b e i n g able to predict a prion the o p t i m u m reac t o r shape. T h i s p r o b l e m also raises interesting c o m p u t a t i o n a l challenges. F i g u r e 5 shows f o u r different shapes c o r r e s p o n d i n g to the grids Illustrated In F i g u r e 2. T h e "champagne bottle" ( F i g u r e 5e) gives the best flow patr tern b u t n o t the o p t i m a l u n i f o r m i t y . T h u s , one has to balance u n i f o r m i t y considerations against recirculations that p r i m a r i l y affect the sharpness o f the Interface In h e t e r o j u n c t l o n g r o w t h .


JENSEN ET AL.

Compound

Semiconductors

and Superlattices

365


19.

F i g u r e 4.

Effect o f Inlet flow rates o n Isotherms, flow streamlines a n d relative d e p o s i t i o n rates o f G a A s ; (a) 70 cc/sec (standard c o n d i t i o n s ) (b) 140 cc/sec (c) 210 c c / s e c . T h e absolute g r o w t h rates scale as 1 ( a ) : 2.6 ( b ) : 3.1 (c).


SUPERCOMPUTER


366

RESEARCH

1 (d)

F i g u r e 5.

Effect o f reactor shapes o n Isotherms, flow streamlines and relative d e p o s i t i o n rates o f G a A s . T h e absolute g r o w t h rates scale as 0.88 ( a ) : 0.91 ( b ) : 0.94 ( c ) : 1.0 ( d ) : 1.08 ( e ) .



19.

JENSEN ET AL.

Compound

Semiconductors

and Superlattices

367

F i g u r e 6 Illustrates the effect o f rotation speed o n flow streamlines a n d the film thickness relative to that at the center o f the susceptor. F o r a stationary susceptor (a) the flow Is e q u i v a l e n t to the b u o y a n c y d o m i n a t e d case In F i g u r e 4 ( a ) . A s the rotation speed Is Increased, the p u m p action o f the susceptor eliminates the b u o y a n c y cell creating a f o r c e d c o n v e c t i o n flow w i t h o u t the disadvantages o f Increasing the flow as was done In F i g ure 4. T h e r o t a t i o n speed also has the advantage o f Increasing the deposit i o n rate. H o w e v e r , the rotation speed can also be too rapid In w h i c h case (cf. F i g u r e 6(c)) flow separation o c c u r along the reactor w a l l w h i c h w i l l lead to b r o a d e n i n g o f Interfaces In the g r o w t h o f h e t e r o j u n c t l o n s . Because o f n o n l i n e a r Interactions b e t w e e n b u o y a n c y , v i s c o u s a n d Inert i a t e r m s m u l t i p l e stable flow fields m a y e x i s t f o r the same parameter v a l u e s as also predicted b y K u s u m o t o et ai ( M ) . T h e bifurcations underl y i n g this p h e n o m e n o n m a y be c o m p u t e d b y the techniques described In the n u m e r i c a l analysis s e c t i o n . The s o l u t i o n structure Is Illustrated In F i g u r e 7 In t e r m s o f the N u s s e l t n u m b e r ( N u , a measure o f the g r o w t h rate) f o r v a r y i n g Inlet flow rate a n d susceptor t e m p e r a t u r e . H e r e the N u s s e l t n u m b e r Is defined as: L(dc/dn)|

s u s c e p t o r

Nu =

(14) c

c

inlet ~ susceptor

F o r Inlet flow rates and susceptor temperatures w i t h i n the cross-hatched area o f F i g u r e 7, two stable flow fields e x i s t c o r r e s p o n d i n g to b u o y a n c y d o m i n a t e d l o w mass transfer and forced c o n v e c t i o n d o m i n a t e d h i g h mass transfer rates. B e t w e e n these stable flow fields there Is an unstable state o n the underside o f the f o l d , where b o t h b u o y a n c y a n d Inertia effects are present. If the susceptor rotation speed Is c h o s e n as a parameter Instead o f the temperature, a s i m i l a r n o n l i n e a r dependence o f N u o n flow rate a n d rotation speed Is o b s e r v e d as Illustrated In F i g u r e 8. A s the rotation speed Is Increased the m u l t i p l i c i t y r e g i o n shifts towards l o w e r Inlet flow rates a n d e v e n t u a l l y v a n i s h e s . T h i s n o n l i n e a r b e h a v i o r m e a n s that reactor o p e r a t i o n c o u l d be d e p e n d e n t o n Its start-up h i s t o r y a n d I m p l i c i t l y o n the operator. F o r e x a m p l e , If the operating c o n d i t i o n s fall w i t h i n the m u l t i p l i city r e g i o n , a r e c i r c u l a t i o n d o m i n a t e d flow a n d c o r r e s p o n d i n g l y l o w N u Is r e a c h e d b y starting the flow once the susceptor Is h o t w h i l e a f o r c e d conv e c t i o n d o m i n a t e d flow a n d c o r r e s p o n d i n g l y h i g h N u Is a c h i e v e d b y establ i s h i n g the flow before heating the susceptor. I n a d d i t i o n , If operation c o n d i t i o n s for h e t e r o j u n c t l o n g r o w t h were c h o s e n near one o f the edge In the f o l d , drastic u n i f o r m i t y a n d g r o w t h rate v a r i a t i o n s c o u l d result. T h u s , these e x a m p l e s demonstrate the advantages o f u s i n g s u p e r c o m p u t l n g to delineate the v a r i a t i o n In system performance w i t h parameters. S i m u l a t i o n o f Sunerlattlce G r o w t h T h e g r o w t h o f superlattices Is one o f the k e y Issues In M O C V D react o r analysis a n d design. I n a d d i t i o n to g r o w i n g h i g h l y u n i f o r m , pure films one m u s t be able to f o r m sharp o r accurately graded Interfaces between


RESEARCH


SUPERCOMPUTER

Effect o f susceptor rotation speeds streamlines a n d relative d e p o s i t i o n zero r p m , (b) 1200 r p m , (c) 2400 g r o w t h rates v a r y as 1.0 ( a ) : 4.0 ( b ) :

o n Isotherms, flow rates o f G a A s ; (a) r p m T h e absolute 4.7 ( c ) .



19. J E N S E N E T A L .

Compound Semiconductors and Superlattices

i.O

F i g u r e 7.

4.5 5.0 5.5 I n l e t Flow (L/mln STP)

6.0

369

β .5

R e d u c e d N u s s e l t n u m b e r f o r mass transfer to the sub strate In a v e r t i c a l reactor f o r v a r y i n g Inlet flow rate a n d susceptor t e m p e r a t u r e .




370

I n l e t Flow ( l / m l n STP)

F i g u r e 8.

R e d u c e d N u s s e l t n u m b e r for mass transfer to the substrate In a vertical reactor for v a r y i n g Inlet flow rate and susceptor r o t a t i o n speed.



19.

JENSEN ET AL.

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Semiconductors

and Superlattices

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s u b s e q u e n t layers to realize a d v a n c e d digital a n d o p t i c a l device designs. T o Illustrate the effect transport p h e n o m e n a o n the sharpness o f the interface b e t w e e n two successive layers, the g r o w t h o f A l A s o n G a A s w i l l be u s e d as a m o d e l . T h i s Is a c o n v e n i e n t e x a m p l e since the transport rates o f A l a n d G a species are s i m i l a r ( 2 Z ) . T o further s i m p l i f y the analysis It a s s u m e d that the s w i t c h i n g system Is perfectly balanced so that the flow Is u n p e r t u r b e d b y the s w i t c h i n g b e t w e e n o r g a n o m e t a l l l c source c o m p o u n d s . T h u s , the steady state flow a n d temperature field m a y be u s e d as Input to the t i m e d e p e n d e n t mass transfer code. T h e set o f o r d i n a r y differential equations r e s u l t i n g f r o m the finite e l e m e n t d i s c r e t i z a t i o n o f the spatial derivatives In E q u a t i o n 7 was Integrated b y a m o d i f i e d v e r s i o n o f E P I S O D E (38) a n d mass l u m p i n g (JJL) was u s e d . U p to 2500 u n k n o w n s were u s e d c o r r e s p o n d i n g to 15 m i n u t e s o f C r a y 2 C P U t i m e . F i g u r e s 9 a n d 10 s h o w the steady state streamlines a n d s i m u l a t e d c o n c e n t r a t i o n t r a n s i t i o n between g r o w t h o f G a A s a n d A l A s f o r two cases, one s t r o n g l y affected b y b u o y a n c y d r i v e n secondary flow ( F i g u r e 9) a n d the o t h e r d o m i n a t e d b y f o r c e d c o n v e c t i o n ( F i g u r e 1 0 ) . A t t i m e zero the a l k y l source Is s w i t c h e d f r o m G a ( C H ) to A 1 ( C H ) . T h e c o l o r scale gives the relative fraction o f the A l - s p e c l e s , I.e. r e d corresponds to pure A l - a l k y l w h i l e v i o l e t corresponds to no A l - a l k y l (pure G a - A l k y l ) . W h e n the flow Is d o m i n a t e d b y b u o y a n c y effects It takes m o r e t h a n 10 seconds before the Ga-specles has b e e n flushed o u t o f the reactor. T h e result Is an Interface w i d e r t h a n 1 0 0 A , w h i c h Is unacceptable f o r m o s t a d v a n c e d d e v i c e s . O n the o t h e r h a n d , w h e n the flow Is f o r c e d c o n v e c t i o n d o m i n a t e d ( F i g u r e 10), the A l - s p e c l e s r a p i d l y reaches the d e p o s i t i o n surface, g i v i n g a reasonably sharp Interface a p p r o x i m a t e l y 6 Â w i d e . H o w e v e r , the r e c i r c u l a t i o n s caused b y the e x p a n s i o n o f the flow cross-section at the Inlet retain the Ga-specles f o r several seconds w h i c h c o u l d cause backg r o u n d d o p i n g . F o r t u n a t e l y , these recirculations can easily be r e m o v e d b y appropriately reshaping the walls as discussed above a n d Illustrated In F i g ure 5. A c o m p a r i s o n between the cases s h o w n In F i g u r e s 9 a n d 10 clearly d e m o n s t r a t e s the d e t r i m e n t a l effect o f recirculations o n Interface abruptness. 3

3

3

3

A l t h o u g h the above predictions have f o c u s e d o n Immediate s w i t c h i n g b e t w e e n o r g a n o m e t a l l l c source c o m p o u n d s , t h e y are also r e l e v a n t to stop g r o w t h procedures, where the flow o f g r o u p III species (e.g. G a ( C H ) ) Is s t o p p e d , the o v e r a l l flow Is balanced, a n d the species Is flushed o u t before the n e w species (e.g. A 1 ( C H ) ) Is I n t r o d u c e d . F o r this procedure It Is critical that the reactor residence t i m e Is k n o w n . T h e present analysis r e a d i l y predicts the time n e e d e d to flush o u t a particular reactant. T h u s , s u p e r c o m p u t e r s i m u l a t i o n c o u l d play a role In d e v e l o p i n g strategies for m a n u f a c t u r i n g superlattlce based electronic a n d p h o t o n i c device structures. 3

3

3

3

Conclusions T h e large scale reactor s i m u l a t i o n s represent a step towards realistic




372

F i g u r e 9.

S t r e a m l i n e s (top) a n d relative gas phase c o m p o s i t i o n o f A l species ( b o t t o m ) In a v e r t i c a l a x l s y m m e t r l c reactor at five different t i m e s d u r i n g g r o w t h o f an A l A s / G a A s superlattlce. R e d corresponds to all A l species, v i o l e t to no A l species. T h e c o r n e r Insert portrays the v a r i a t i o n i n s o l i d fraction o f A l across the interface. B u o y a n c y d o m i n a t e d flow.



F i g u r e 10.

S t r e a m l i n e s (top) and relative gas phase c o m p o s i t i o n o f A l species ( b o t t o m ) In a v e r t i c a l a x l s y m m e t r l c reactor at five different times d u r i n g g r o w t h o f an A l A s / G a A s superlattlce. R e d corresponds to all A l species, v i o l e t to no A l species. The c o r n e r Insert portrays the v a r i a t i o n In s o l i d fraction o f A l across the Interface. F o r c e d c o n v e c t i o n d o m i n a t e d flow.



374


CVD reactor descriptions and away from oversimplified boundary layer and film theory models. The model predictions offlow,temperature, and concentration fields In horizontal reactors demonstrate that threedimensional effects may be Important even In the absence of buoyancy effects. Furthermore, the presence of buoyancy driven rolls greatly enhance growth rate variations. The vertical reactor analyses show that the film thickness uniformity Is strongly Influenced by susceptor edge, reactor wall, and buoyancy effects. Furthermore, nonlinear Interactions between transport processes may lead to abruptflowtransitions with detrimental effects on growth performance. The simulation of superlattlce growth Illustrates that realization of sharp heterojunctlon Interfaces require thatflowrecirculations and eddies be avoided. However, additional modelling efforts are needed to relax assumptions and further develop realistic models. The nature of the three-dimensionalflowfield needs to be understood. The treatment of surface reactions In current models Is limited, In particular with respect to the Initial nucleatlon and growth of the solid film. Further development of supercomputers and numerical methods In conjunction with experimental Investigation are needed to formulate accurate models linking process conditions (flow, concentration, temperature) to microscopic material properties such as film crystal structure and defect concentrations. Acknowledgments The research forming the basis for this paper was supported by the National Science Foundation, by the Minnesota Supercomputer Institute, by the University of Minnesota Microelectronics and Information Sciences Center, and by a Teacher-Scholar Award from the Camille and Henry D reyfus Foundation to KFJ. Literature Cited 1. Ludowise, M. J. J. Appl. Phys. 1985, 58, R31. 2. Dupuis, R. D. Science 1984, 226, 623. 3. Kuech, T. F.; Veuhoff, E.; Kuan, T. S.; Deline, V.; Potemski, R. J. Crystal Growth 1986, 77, 257. 4. Giling, L. J. J. Electrochem. Soc. 1982, 129, 634. 5. Van de Ven, J.; Rutten, G. M. J.; Raaijmakers, M. J.; Giling, L. J. J.Crystal Growth 1986, 76, 352. 6. Ban, V. S. J. Electrochem. Soc. 1978, 125, 317. 7. Eversteijn, F. C.; Peek, H. L. Phillips Res. Rep. 1972, 25, 472. 8. Takahashi, R.; Koza Y.; Sugawara, K. J. Electrochem. Soc. 1972, 119, 1406. 9. Stock, L.; Richter, W. J. Crystal Growth 1986, 77, 128. 10. Talbot, L.; Cheng, R. K.; Scheffer, R. W.; Willis, J. Fluid Mech. 1980, 101, 737.


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11.

Wang, C. Α.; Groves, S. H.; Palmteer, S. C.; Weyburne, D. W.; Brown, R. A. J. Crystal Growth 1986, 77, 136. 12. Wahl, G. Thin Solid Films 1977, 40, 13. 13. Williams, J. E.; Peterson, R. W. J. Crystal Growth 1986, 77, 128. 14. Bird, R. B.; Stewart, W. E.; Lightfoot, Ε. N. Transport Phenomena, Wiley: New York, 1960. 15. Jenkinsen, J. P.; Pollard, R. J. Electrochem. Soc. 1984, 131, 425. 16. Moffat, H.; Jensen, K. F. J. Electrochem. Soc. (submitted). 17. Coltrin, M. E.; Kee, R. J.; Miller, J. A. J. Electrochem. Soc. 1986, 133, 1206. 18. Reep, D. H.; Ghandi, S. K. J. Electrochem. Soc. 1983, 130, 675. 19. Zienkiewicz, O. C. The Finite Element Method, Third Edition, McGraw-Hill (UK): London, 1983. 20. Huyakom, P.; Taylor, C.; Lee, R.; Gresho, P. Computers and Fluids 1978,6,25. 21. Petzold, L. R. Sandia National Laboratories Report, SAND82-8637, Sept. 1982. 22. Houtman, C.; Graves, D. B.; Jensen, K. J. Electrochem. Soc. 1986, 133, 961. 23. Kieda, S.; Fotiadis, D. I.; Jensen, K. F. Crystal Growth (in prepara tion). 24. Fotiadis, D. I.; Jensen K. F. Int. J. Num. Meth. Fluids (in prepara tion). 25. Ryskin, G.; Leal, L. G. J. Comp. Phys. 1983, 50, 71. 26. Keller, Η. B. "Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems," in Applications of Bifurcation Theory Rabinowitz,P., Ed.; Academic Press: New York, 1977. 27. Keller, H. B. SIAM J. Sci. and Stat. Comp. 1983, 4 573. 28. Chan, T. F. SIAM J. Sci. and Stat. Comp. 1984, 5, 135. 29. Hess, D. W.; Jensen, K. F.; Anderson, T. J. Reviews in Chem. Engn. 1985, 3, 97. 30. Juza, J.; Cermak, J. J. Electrochem. Soc. 1982, 129, 1627. 31. Coltrin, M. E.; Kee, R. J.; Miller, J. A. J. Electrochem. Soc. 1984, 131, 425. 32. Brelland, W. G.; Coltrin, M. E.; Ho, P. J. Appl. Phys. 1986, 59, 3267. 33. Moffat, H.; Jensen, K. F. J. Crystal Growth 1986, 77, 108. 34. Chiu, K.C.;Rosenberg, F. Int. J. Heat and Mass Transfer (in pres 35. Schlichting, H. Boundary Layer Theory, McGraw-Hill: New York, 1979. 36. Kusumoto, Y.; Hayashi, T.; Komiya, S. Japan J. Appl. Phys. 1985, 24, 620. 37. Suzuki, M.; Sato, M. J. Electrochem. Soc. 1985, 132, 1684. 38. Byrne, G.; Hindmarsh, A. ACM Transcations on Mathematica Software 1975, 1, 71. RECEIVED June 15, 1987 Jensen and Truhlar; Supercomputer Research in Chemistry and Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Growth of Compound Semiconductors and Superlattices by

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