Liquid-Phase Epitaxy and Phase Diagrams of Compound

3 Liquid-Phase Epitaxy and Phase

Downloaded by UNIV OF PITTSBURGH on May 14, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1989-0221.ch003

Diagrams of Compound Semiconductors

Timothy J. Anderson Department of Chemical Engineering, University of F l o r i d a , Gainesville, FL 32611

The fundamentals of liquid-phase epitaxy of compound semiconductors are presented. Selected topics associated with the chemical processing of semiconductors by this technique are discussed. These topics include compositional variations, growth mechanisms, and the initial stages of epitaxy. A formalism for the treatment of solid-liquid phase equilibrium in multicomponent compound-semiconductor systems is given. Procedures for interpolating, extrapolating, and predicting multicomponent phase diagrams are suggested.

E P I T A X Y IS T H E P R O C E S S O F G R O W I N G r e g u l a r l y o r i e n t e d , t h i n films o n a substrate. T h e goals of epitaxy are to p r o d u c e perfectly crystalline films w i t h c o n t r o l of c o m p o s i t i o n , thickness, a n d p u r i t y a n d to r e p r o d u c i b l y repeat the process o n e i t h e r a n e w substrate or a p r e v i o u s l y d e p o s i t e d

film.

Several

processes for epitaxial film g r o w t h are available, i n c l u d i n g l i q u i d - p h a s e e p itaxy ( L P E ) , c h e m i c a l v a p o r d e p o s i t i o n ( C V D ) , m o l e c u l a r - b e a m

epitaxy

( M B E ) , plasma deposition, and ion-cluster-beam deposition. These deposition technologies are at different stages of d e v e l o p m e n t a n d offer c e r t a i n or p o t e n t i a l advantages for the fabrication of solid-state devices.

Advantages and Limitations of Liquid-Fhase Epitaxy L i q u i d - p h a s e epitaxy is the process o f g r o w i n g films from a l i q u i d s o l u t i o n a n d is an attractive m e t h o d o f film g r o w t h for several reasons. L P E was the 0065-2393/89/0221-0105$15.9070 © 1989 American Chemical Society

Hess and Jensen; Microelectronics Processing Advances in Chemistry; American Chemical Society: Washington, DC, 1989.

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MICROELECTRONICS PROCESSING: C H E M I C A L ENGINEERING ASPECTS


first process b r o a d l y d e v e l o p e d for the g r o w t h of c o m p o u n d - s e m i c o n d u c t o r materials a n d has p r o v e n to be a reliable process. C o m p a r e d w i t h other methods, L P E is a s i m p l e process and thus requires a short d o w n t i m e a n d demands less operator s k i l l . T h e r e q u i r e d i n v e s t m e n t i n capital e q u i p m e n t is modest, a n d the operating expenses are m i n i m a l relative to other g r o w t h technologies. I n L P E , film g r o w t h occurs v e r y near the e q u i l i b r i u m state, a n d thus the t e c h n i q u e is r e p r o d u c i b l e a n d gives films w i t h l o w concentrations of g r o w t h - i n d u c e d defects. T h e starting materials are generally i n e l e m e n t a l f o r m , a n d t h e i r availability i n h i g h p u r i t y translates into l o w levels of u n i n t e n t i o n a l d o p i n g . I m p u r i t y levels are further decreased b y the t e n d e n c y of most contaminants to segregate into the m e l t . F u r t h e r m o r e , the l i q u i d metal solvents u s e d i n L P E can dissolve most elements to give a large selection of dopants. T h e h i g h g r o w t h rate that can b e a c h i e v e d b y L P E reduces g r o w t h t i m e for t h i c k - f i l m applications. I n a d d i t i o n , processing from the m e l t tends to equalize the d i s t r i b u t i o n coefficients o f the l i q u i d constituents. I n contrast, large differences i n d i s t r i b u t i o n coefficients can exist b e t w e e n the transportable vapor species u s e d i n other processes. F o r example, s o l i d solutions c o n t a i n i n g A l m i x e d w i t h G a or I n are easy to grow b y L P E b u t are difficult to g r o w w i t h n e a r - e q u i l i b r i u m C V D techniques. R e l a t i v e to some alternative g r o w t h methods, safety considerations for L P E are less r e s t r i c t i v e , w i t h h y d r o g e n h a n d l i n g b e i n g the most hazardous operation. T h e success of L P E is e v i d e n t from its p r o m i n e n c e i n the p r o d u c t i o n of c o m m e r c i a l devices, particularly for optoelectronic applications. F o r m a n y device structures, the highest device performance characteristics have b e e n a c h i e v e d i n devices produced by L P E . A l t h o u g h the advantages of L P E are considerable, this process has several serious drawbacks. L P E is i n h e r e n t l y a batch process a n d is usually scaled u p b y the s i m p l e d u p l i c a t i o n of systems. Single-wafer batch processing a n d l o n g b a k i n g times a l l o w l o w wafer t h r o u g h p u t c o m p a r e d w i t h the t h r o u g h p u t r e a l i z e d w i t h S i C V D technology. G r e a t care must be taken to y i e l d acceptable surface morphologies, because the process is often p l a g u e d b y surface defects such as i n c o m p l e t e m e l t r e m o v a l , terrace formation, p i n holes, a n d meniscus lines. T h i c k n e s s u n i f o r m i t y can be p o o r as a result of natural a n d forced convection i n the m e l t a n d variations i n substrate t e m perature. Because of h i g h i n i t i a l g r o w t h rates, film thicknesses of less t h a n several h u n d r e d angstroms or w i t h i n tight thickness tolerances are e x t r e m e l y difficult to achieve. D u r i n g the g r o w t h of solid solutions (e.g., p s e u d o b i n a r y alloys a n d d o p e d compounds), the m e l t c o m p o s i t i o n at the interface can be a f u n c t i o n of t i m e , a situation that results i n a natural c o m p o s i t i o n gradient i n the g r o w t h d i rection. H e t e r o e p i t a x y , the g r o w t h of an epitaxial film w i t h a c o m p o s i t i o n different from that of the substrate, is m o r e difficult c o m p a r e d w i t h other


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107

techniques because of m e l t b a c k p r o b l e m s . F o r example, it is difficult to g r o w a layer of I n P o n I ^ G a ^ ^ s w i t h an abrupt interface because of the l i m i t e d transport of As a n d G a across the interface. D e s p i t e these limitations, l i q u i d phase epitaxy is still c o m m o n l y u s e d to grow epitaxial films of c o m p o u n d semiconductors. S e v e r a l excellent reviews of l i q u i d - p h a s e epitaxy have a p p e a r e d i n the literature over the past 15 years (1-12). T h e discussion i n this chapter w i l l be l i m i t e d i n scope b u t w i l l s u p p l e m e n t the m a t e r i a l discussed i n p r e v i o u s reviews. I n p a r t i c u l a r , issues that can be a n a l y z e d b y t r a d i t i o n a l methods o f


c h e m i c a l e n g i n e e r i n g are addressed for this c h e m i c a l process. Because the g r o w i n g s o l i d - l i q u i d interface is near e q u i l i b r i u m , the calculation of m u l t i c o m p o n e n t c o m p o u n d - s e m i c o n d u c t o r phase diagrams w i l l b e e m p h a s i z e d .

LPE Growth Techniques T h e L P E g r o w t h system must p r o v i d e a d r i v i n g force for n u c l e a t i o n of the solid film, a p h y s i c a l means of contacting a l i q u i d solution to the substrate, an efficient way of r e m o v i n g the solution from the substrate after g r o w t h , and, for most device applications, a m e t h o d of r e p e a t i n g the g r o w t h process from a solution of different c o m p o s i t i o n .

Nucleation. T h e d r i v i n g force must decrease the G i b b s energy of the l i q u i d solution near the substrate to a value sufficiently l o w to p r o d u c e n u c l e a t i o n o n the substrate b u t not so l o w that homogeneous or heterogeneous n u c l e a t i o n o n nonsubstrate s o l i d surfaces occurs. T h u s a s m a l l w i n d o w i n the values of the solution G i b b s energy exists i n w h i c h successful g r o w t h can take place. T h e u p p e r l i m i t of the G i b b s energy is d e t e r m i n e d p r i m a r i l y b y substrate properties (i.e., orientation, i m p u r i t y content, c r y s t a l l i n i t y , a n d c o m p o s i tion), whereas the l o w e r l i m i t is fixed b y the properties of the l i q u i d solution and the materials of construction. T h e solution G i b b s e n e r g y is a function of t e m p e r a t u r e , of c o m p o s i t i o n , a n d , w e a k l y , of pressure. M o s t L P E t e c h niques are d r i v e n b y t e m p e r a t u r e decreases caused b y p r e c o o l i n g of the substrate, continuous or step c o o l i n g of the solution, P e l t i e r cooling, or the establishment o f a t e m p e r a t u r e gradient b e t w e e n the substrate a n d a c o n stant-composition source. E l e c t r o e p i t a x y , i n w h i c h a large c u r r e n t is passed n o r m a l to the substrate, represents an example i n w h i c h the l i q u i d c o m position near the s o l i d - l i q u i d interface is changed b y e l e c t r o m i g r a t i o n , i n a d d i t i o n to P e l t i e r cooling. I n response to a d r i v i n g force, the system adjusts the value of the l i q u i d G i b b s e n e r g y b y c h a n g i n g the c o m p o s i t i o n t h r o u g h d e p o s i t i o n . I n heteroepitaxy, the r e q u i r e d d r i v i n g force d u r i n g the i n i t i a l stages of epitaxy is clearly different from that r e q u i r e d d u r i n g subsequent g r o w t h , w h i c h is essentially h o m o e p i t a x i a l deposition.


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MICROELECTRONICS PROCESSING: C H E M I C A L E N G I N E E R I N G ASPECTS

Contact between Solution and Substrate. T h e p h y s i c a l processes u s e d to contact the l i q u i d solution to the substrate can be classified b r o a d l y into t i p p i n g , d i p p i n g , or s l i d i n g methods ( F i g u r e 1).


A t i p p i n g p r o c e d u r e was first suggested b y N e l s o n (13). I n this m e t h o d , a l i q u i d solution is successfully b r o u g h t into a n d r e m o v e d f r o m contact w i t h the substrate b y e i t h e r t i p p i n g a h o r i z o n t a l boat ( F i g u r e la) or rotating a c y l i n d e r ( F i g u r e l b ) c o n t a i n i n g a fixed substrate a n d the solution. T i p p i n g techniques are s i m p l e , b u t m e l t r e m o v a l is difficult i f the t e c h n i q u e relies o n gravity. T h e early versions of these designs c o u l d not g r o w m u l t i p l e layers a n d gave p o o r thickness u n i f o r m i t y . B o t h h o r i z o n t a l ( F i g u r e l c ) a n d v e r t i c a l ( F i g u r e Id) d i p p i n g processes have b e e n investigated, b u t again, these techniques r e l y o n gravity for m e l t r e m o v a l . T o i m p r o v e thickness u n i f o r m i t y , d i p p i n g techniques are often accompanied b y m e c h a n i c a l s t i r r i n g . I n m o d e r n L P E systems, film g r o w t h is almost exclusively p e r f o r m e d i n a l i n e a r sliding-boat design ( F i g u r e l e ) . T h e sliding-boat t e c h n i q u e has e v o l v e d o v e r the past 20 years, w i t h m a n y variations existing i n different laboratories. T h e basic design consists of t w o m a i n pieces ( F i g u r e l e ) : a

THERMOCOUPLE

SUBSTRATE

« 1

SOLUTION

GRAPHITE BOAT

Figure la. LPE growth systems: tipping technique. (Reproduced with permission from reference 13. Copyright 1963 RCA Research and Engineering.)

SUBSTRATE CLAMP SOLUTION Figure lb. LPE growth systems: rotary method.


ANDERSON

Liquid-Phase

Epitaxy


SUBSTRATE

Figure lc. LPE growth systems: horizontal dipping method.

1 1 1

RADIATION SHIELDS

Figure Id. LPE growth systems: vertical dipping method.


110



SLIDE Figure le. LPE growth systems: multibin sliding-boat method. s l i d e r plate w i t h a substrate recess a n d a m u l t i p l e - w e l l assembly. A substrate fitted i n the s l i d e r plate recess can m o v e relative to the w e l l assembly to place the substrate b e n e a t h a solution w e l l . T h e pieces a n d recess are m a c h i n e d to fine tolerances to a l l o w m e c h a n i c a l w i p i n g of the m e l t f r o m the film after g r o w t h . T h e use of m u l t i p l e wells p e r m i t s the g r o w t h of m u l t i p l e layers o n a single substrate or the p r o d u c t i o n of m u l t i p l e substrates. T h e s l i d e r boat is usually constructed of graphite, although alternative materials such as b o r o n n i t r i d e a n d silica (14) have b e e n used. G r a p h i t e is readily o b t a i n e d i n h i g h p u r i t y , easily m a c h i n e d , relatively i n e r t w i t h respect to the solution, a n d n e a r l y frictionless i n operation. G r a p h i t e boats do have a l i m i t e d l i f e t i m e , a n d care m u s t be exercised i n d e s o r b i n g gases contained i n this r e l a t i v e l y porous m a t e r i a l . T h e s l i d e r boat assembly is n o r m a l l y s u r r o u n d e d b y a fused silica t u b e . R e d u c t i o n of the silica tube b y h y d r o g e n can u n i n t e n t i o n a l l y dope the s e m i c o n d u c t o r w i t h S i (15), a n d the a d d i t i o n of s m a l l amounts of C I (as m e t a l c h l o r i d e or HC1) can greatly reduce this p r o b l e m (16). T h e slider boat is b l a n k e t e d i n a flowing stream of palladium-alloy-diffused h y d r o g e n . E x t r e m e care m u s t be u s e d to m i n i m i z e the l e v e l of oxygen i n the system, p a r t i c u l a r l y for the g r o w t h of films c o n t a i n i n g A l . L o w oxygen levels are a c h i e v e d b y p r e b a k i n g the system, i m p l e m e n t i n g good h o u s e k e e p i n g p r o c e d u r e s , a n d u s i n g p u r i f i e d h y d r o g e n . M a n y systems are e q u i p p e d w i t h a h y g r o m e t e r to measure H 0 levels. T h e gas stream m a y also contain d e s i r e d dopants (17) or major species u s e d to r e d u c e the evaporation of volatile components (18). T h e arrangement is usually p l a c e d i n a resistance-heated furnace, a n d t e m perature c o n t r o l to better than 0.1 °C is r e q u i r e d to achieve r e p r o d u c i b l e g r o w t h thicknesses. T o f u r t h e r i m p r o v e r e p r o d u c i b i l i t y , m o d e r n systems are generally e q u i p p e d w i t h automatic s l i d e r p u l l i n g - a n d - p u s h i n g mechanisms a n d automatic t e m p e r a t u r e p r o g r a m m e r s . 2

Solvent Selection. T h e characteristics of a good solvent i n c l u d e l o w vapor pressure ( and Δ Η / reference values were obtained from references 123 (Δ and O ) , 137 (Θ), and 138 (•). InS

u

of the s t o i c h i o m e t r i c l i q u i d . W i t h m e t h o d I , t h e activity coefficients m u s t be calculated at each l i q u i d u s t e m p e r a t u r e T\ whereas w i t h m e t h o d I I I , o n l y the activity coefficients at the m e l t i n g p o i n t of the c o m p o u n d are r e quired. T h e most c o m m o n l y u s e d solution m o d e l to represent t h e l i q u i d - p h a s e behavior is t h e s i m p l e - s o l u t i o n m o d e l . F i g u r e 18 shows a c o m p a r i s o n b e t w e e n the r e c o m m e n d e d value of 6 j a n d the value calculated from t h e A

s b


3.

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ANDERSON

Epitaxy T(K)


Tm700

14.0

600

500

18.0

400

22.0

300

26.0

30.0

34.0

38.0

10 /T(K) 4

Figure 17. (c)

Qi»sb : m

— , upper and lower limits; and —, recommended value.

parameters of the s i m p l e - s o l u t i o n m o d e l (142-145). T h e s e parameters w e r e estimated from a fit to the l i q u i d u s measurements. I n a l l eases, a reasonable fit of the l i q u i d u s data was r e p o r t e d , but the calculated values of Q show significant variation ( F i g u r e 18). O b v i o u s l y variations i n the l i q u i d - s o l u t i o n activities must exist to y i e l d a reasonable fit to the phase d i a g r a m , a n d these variations b e c o m e m o r e p r o n o u n c e d as temperature is decreased. T h u s , an accurate d e s c r i p t i o n of the l i q u i d - s o l u t i o n behavior is difficult to obtain b y A]sh


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MICROELECTRONICS PROCESSING: C H E M I C A L E N G I N E E R I N G ASPECTS

Table Π. Values of θ** * A ,* τ (Κ) 1


v

Reference

0.3

774.3

-3.41

125

0.4

788.6

-3.12

125

0.5

798.2

-3.01

125

0.6

787.6

-3.14

125

0.7

756.9

-3.56

125

0.8

699.3

-4.42

125

0.9

607.2

-6.24

125

0.3

774.3

-2.87

141

0.4

788.6

-2.69

141

0.5

798.2

-2.55

141

0.6

787.6

-2.66

141

0.7

756.9

-3.04

141

0.8

699.3

-3.87

141

0.9

607.2

-5.61

141

0.398

788.3

-2.90

116

0.500

798.2

-2.60

116

0.600

787.6

-2.75

116

0.656

772.1

-2.78

116

0.703

755.8

-3.16

116

0.787

707.8

-3.96

116

0.890

616.7

-5.81

116

u s i n g m e t h o d I c o m b i n e d w i t h phase d i a g r a m data to estimate l i q u i d - s o l u t i o n - m o d e l parameters. T o test t h e effect o f u s i n g a solution m o d e l i n t h e calculation o f Θ t h e s i m p l e - s o l u t i o n m o d e l was u s e d i n c o n j u n c t i o n w i t h e i t h e r m e t h o d I o r m e t h o d I I I to fit sets o f data consisting o f the l i q u i d u s t e m p e r a t u r e alone (146), l i q u i d u s t e m p e r a t u r e a n d e n t h a l p y o f m i x i n g (147), l i q u i d u s t e m p e r ature a n d activity (147), a n d a l l three types o f data c o m b i n e d . W i t h t h e parameters d e t e r m i n e d from t h e fit, values of θ y as a function of t e m p e r a t u r e w e r e calculated a n d c o m p a r e d w i t h t h e r e c o m m e n d e d value. Μ

F i g u r e 19 shows t h e results o b t a i n e d w i t h m e t h o d I for t h e A l - S b system. E x a m i n a t i o n o f t h e results for t h e fit o f t h e l i q u i d u s t e m p e r a t u r e alone (the usual procedure) indicates that b o t h t h e activity p r o d u c t at t h e m e l t i n g t e m p e r a t u r e a n d t h e t e m p e r a t u r e d e p e n d e n c e o f Q^sb repre sented rather poorly. T h e results for the data base consisting o f the l i q u i d u s a n d t h e e n t h a l p y o f m i x i n g are significantly better than t h e results o b t a i n e d w i t h t h e l i q u i d u s data set alone. O n l y w h e n activity data are also i n c l u d e d i n t h e data set does t h e v a l u e o f Q agree w i t h t h e r e c o m m e n d e d v a l u e . a

r

e

mh

T h e results o f s i m i l a r calculations u s i n g the same solution data base a n d m e t h o d I I I are s h o w n i n F i g u r e 20. I n a l l cases, t h e agreement b e t w e e n the calculated value o f a n d t h e r e c o m m e n d e d value is i m p r o v e d . A s i m i l a r i m p r o v e m e n t b y u s i n g m e t h o d I I I has also b e e n s h o w n for the G a - S b system (148). F o r those systems i n w h i c h n o solution measurements have


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Liquid-Phase Epitaxy

147

τ(κ)


Tm 1000

700

10.0

15.0

500

400

20.0

25.0

300

30.0

35.0

ιο /τ(κ) 4

Figure 18. Values of Q i b calculated from solution model parameters esti mated by fitting the phase diagram. Data are from references 144 ( ), 143 (···)» 142 (— · — ) , and 145 (—). The recommended values are indi cated by —. A S

l

b e e n r e p o r t e d (e.g., phosphides), m e t h o d I I I is a superior p r o c e d u r e f o r estimating a n d solution m o d e l parameters. B y u s i n g the procedures j u s t o u t l i n e d , the r e d u c e d standard-state c h e m ical potential can b e estimated for a l l compounds. T h i s value o f θ y is v a l i d for any s o l i d - l i q u i d phase e q u i l i b r i u m p r o b l e m that contains the c o m p o u n d

American Chemical Society library 1155 16th St.. Hess and Jensen; Microelectronics Processing Advances in Chemistry;Washington, American Chemical Society: 20036 Washington, DC, 1989.



148

Figure 19. θ W as a function of reciprocal temperature. Values were calculated by the simple-solution model, with parameters estimated from a fit of the combined data set ( · · · ), liquidus data only (---), liquidus and activity data ( ), and liquidus and enthalpy of mixing data (— ). The recommended values are indicated by —.


3.

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Liquid-Phase


Tm

-Q

-10.0

h

-15.0

h

-20.0

h

-25.0

h

-30.0

h

149

Epitaxy

1000

10.0

15.0

20.0

25.0

30.0

35.0

ιο7τ(κ) Figure 20. Values of %Aisb versus reciprocal temperature. Values were calculated with simple-solution model parameters estimated by a fit to the combined data set (— —) and nearly coincident with the recommended values, liquidus data only (—), liquidus and activity data (— · —), and liq uidus and enthalpy of mixing data ( · · · ). The recommended values are indi cated by —. m


150


(pure o r i n solution). H o w e v e r , t h e available data base for some systems is not c o m p l e t e (e.g., arsenides a n d phosphides). T h e lack o f a c o m p r e h e n s i v e data base is p r i m a r i l y a result o f difficulties i n p e r f o r m i n g t h e r e q u i r e d e x p e r i m e n t s (e.g., the h i g h v a p o r pressures o f the arsenides a n d phosphides). Because the standard state is e n t i r e l y arbitrary, an i n f i n i t e - d i l u t i o n standard state w o u l d a v o i d the e x p e r i m e n t a l difficulties e n c o u n t e r e d w i t h h i g h - v a p o r pressure species. U n f o r t u n a t e l y , t h e i n f i n i t e - d i l u t i o n properties o f these species have r e c e i v e d little e x p e r i m e n t a l attention. I n t h e f o r m u l a t i o n o f the phase e q u i l i b r i u m

Determination of Γ^·. Downloaded by UNIV OF PITTSBURGH on May 14, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1989-0221.ch003

p r o b l e m p r e s e n t e d e a r l i e r , c o m p o n e n t c h e m i c a l potentials w e r e separated i n t o three t e r m s : (1) θ^, w h i c h expresses the. p r i m a r y t e m p e r a t u r e d e p e n d e n c e , (2) solution m o l e fractions, w h i c h represent the p r i m a r y c o m p o s i t i o n d e p e n d e n c e (ideal entropie contribution), a n d (3) Γ^, w h i c h accounts for relative m i x t u r e nonidealities. Because little data about t h e e x p e r i m e n t a l properties o f solutions exist, Γ is usually evaluated b y i m p o s i n g a m o d e l to Μ

d e s c r i b e t h e b e h a v i o r o f the l i q u i d a n d s o l i d mixtures a n d e s t i m a t i n g m o d e l parameters b y s e m i e m p i r i c a l methods o r fitting l i m i t e d segments o f t h e phase d i a g r a m . Various solution models u s e d to describe the l i q u i d a n d s o l i d mixtures are discussed i n t h e f o l l o w i n g sections, a n d t h e b e h a v i o r o f

is

presented. Liquid-Solution

T h e s i m p l e - s o l u t i o n m o d e l has b e e n

Modeh.

used

most extensively to describe t h e d e p e n d e n c e o f the excess i n t e g r a l m o l a r G i b b s energy, G

x s

, o n t e m p e r a t u r e a n d c o m p o s i t i o n i n b i n a r y (142-144,

149-155), quasi b i n a r y (156-160), ternary (156, 160-174), a n d q u a t e r n a r y (175-181) c o m p o u n d - s e m i c o n d u c t o r phase d i a g r a m calculations. F o r a s i m p l e m u l t i c o m p o n e n t system, t h e excess i n t e g r a l molar G i b b s energy o f so l u t i o n is expressed b y

G

M

=

£Σ Σ

« W i

(«)

* k=l j=l;j#k i n w h i c h t h e interchange energies are e q u a l (w^ = w^, are functions o f t e m p e r a t u r e a n d pressure, a n d are i n d e p e n d e n t o f composition. F o r c o n d e n s e d phases, t h e pressure d e p e n d e n c e o f can b e n e g l e c t e d a n d u> is usually p e r m i t t e d a l i n e a r t e m p e r a t u r e d e p e n d e n c e , a + bT ( i n w h i c h a a n d b are constants). A strictly regular solution (with r a n d o m m i x i n g a n d excess e n t r o p y o f m i x i n g [ S ] is 0) a n d a n a t h e r m a l solution (enthalpy o f m i x i n g [ Δ Η ] is 0) are t w o l i m i t i n g cases o f the s i m p l e solution. F o r strictly regular solutions, u ; = a, a n d deviations f r o m ideal-solution b e h a v i o r arise from heat effects, whereas for a t h e r m a l solutions, w = bT, a n d deviations from ideality arise from e n t r o p y rather t h a n heat effects. kj

xs

mix

k j

ki


3.

ANDERSON

Liquid-Phase

151

Epitaxy

I n general, the s i m p l e - s o l u t i o n m o d e l performs q u i t e w e l l for s y m m e t r i c b i n a r y systems (e.g., G a - S b ) . Inspection o f equation 4 3 w i t h η = 2 shows that t h e excess i n t e g r a l molar G i b b s energy is a s y m m e t r i c a l function o f c o m p o s i t i o n for b i n a r y systems. F o r a h i g h l y a s y m m e t r i c system (e.g., A l - S b ) , A n d e r s o n et a l . (143) a n d Joullie a n d G a u t i e r (144) u s e d different values o f the i n t e r a c t i o n p a r a m e t e r o n e i t h e r side o f the c o m p o u n d m e l t i n g p o i n t , whereas C h e n g et a l . (182) a d d e d a concentration-dependent t e r m to the i n t e r a c t i o n p a r a m e t e r t# . S i m i l a r l y , a c o m p o s i t i o n - d e p e n d e n t w has b e e n u s e d to describe t h e G a - I n (183), G a - A s , a n d G a - P systems (157). T h e use of two different values of w for a b i n a r y c o m p o u n d i n the p r e d i c t i o n of a t e r n a r y phase diagram w i l l give a d i s c o n t i n u i t y i n t h e l i q u i d u s at t h e quasi b i n a r y c o m p o s i t i o n . I n a d d i t i o n , the use o f this extra t e r m to calculate a m u l t i c o m p o n e n t phase d i a g r a m w i l l l e a d to a t h e r m o d y n a m i c inconsistency i n expressions o f ternary a c t i v i t y , because t h e G i b b s - D u h e m e q u a t i o n is not satisfied. i$

kj


ki

A l t h o u g h t h e s i m p l e - s o l u t i o n m o d e l provides a good analytical r e p r e sentation o f the b i n a r y phase diagrams, good values for the t h e r m o d y n a m i c properties o f the l i q u i d solution are not o b t a i n e d w h e n parameters d e t e r m i n e d f r o m a fit o f the b i n a r y l i q u i d u s are used. F o r example, values o f the e n t h a l p y o f m i x i n g p r e d i c t e d f r o m these l i q u i d u s fits are always p o s i t i v e , whereas available e x p e r i m e n t a l data show negative values. I n d e e d , t h e e n thalpy o f m i x i n g is e x p e c t e d to b e always negative because o f t h e strong attractive interactions. T h i s expectation is also expressed i n t h e phase d i a grams as a negative d e v i a t i o n from i d e a l i t y a n d a t e n d e n c y t o w a r d c o m p o u n d formation. S i m i l a r l y , T h u r m o n d (150) a n d A r t h u r (15J) found that t h e i n t e r a c t i o n coefficients o b t a i n e d f r o m a fit o f the e x p e r i m e n t a l l i q u i d u s o r vapor pressure i n the arsenide a n d p h o s p h i d e systems d i d not p r o d u c e the same t e m p e r a t u r e d e p e n d e n c e . P a n i s h et a l . (142, 154) p o i n t e d out that these discrepancies may b e d u e to (1) errors r e s u l t i n g from the assumed values for A H / a n d t h e approximation A C [ i j ] = 0 i n θ^ , (2) deviations from s i m p l e - s o l u t i o n b e havior, o r (3) uncertainties i n t h e i n t e r p r e t a t i o n o f the vapor pressure data, because some o f the quantities necessary i n the calculations are not accurately k n o w n (e.g., reference-state vapor pressures for p u r e l i q u i d A s a n d P ) . K n o b l o c h et a l . (184, 185) a n d P e u s c h e l et a l . (186, 187) have o b t a i n e d excellent agreement b e t w e e n calculated a n d e x p e r i m e n t a l activities a n d v a p o r pressures w i t h t h e use o f K r u p k o w s k i ' s a s y m m e t r i c a l f o r m a l i s m for ac t i v i t y coefficients, whereas Ilegems et al. (Ill) d e m o n s t r a t e d that satisfactory agreement b e t w e e n l i q u i d u s a n d vapor pressure measurements exists w h e n an accurate expression for t h e l i q u i d u s is u s e d . j

p

1

I n a d d i t i o n , several other models have b e e n used w i t h m e t h o d I to calculate b i n a r y o r ternary phase diagrams (183, 188-201). A m o n g these models are the quasi c h e m i c a l e q u i l i b r i u m m o d e l (188,190), t r u n c a t e d M a r gules expansions (183,191, 192), G a u s s i a n f o r m a l i s m (193), orthogonal series


152


expansions (194), D a r k e n ' s f o r m a l i s m (195), a n d various c h e m i c a l theories (196-201). T h e most p o p u l a r c h e m i c a l theory postulates stoichiometric c h e m i c a l species that interact accordingly as a regular solution (regular associatedsolution model). T h e associated-solution m o d e l is based usually o n the f o l l o w i n g assumptions: 1. the m o l e c u l e l i k e stoichiometric species of u n l i k e atoms c a l l e d " c l u s t e r s " , "associates", o r " c o m p l e x e s "

exist i n the l i q u i d


state, 2. the associated complexes are i n a d y n a m i c e q u i l i b r i u m w i t h the nonassociated atoms that can b e d e s c r i b e d b y a mass action law, 3. the associated complexes behave as i n d e p e n d e n t particles, 4. a l l species are statistically d i s t r i b u t e d , a n d 5. the excess t h e r m o d y n a m i c properties consist of contributions from b o t h the p h y s i c a l interactions a n d the c h e m i c a l reactions. T h i s m o d e l has b e e n w i d e l y u s e d for strongly i n t e r a c t i n g systems that e x h i b i t a s y m m e t r i c properties. H o w e v e r , the p r o b l e m becomes c o m p l i c a t e d w h e n this m o d e l is g e n e r a l i z e d to m u l t i c o m p o n e n t systems because of the possib i l i t y of n e w species b e i n g f o r m e d . T h e formation of n e w species results i n a large n u m b e r of parameters i n v o l v e d i n the calculation. T h e case of b i n a r y s o l i d - l i q u i d e q u i l i b r i u m p e r m i t s one to focus o n l i q u i d - p h a s e nonidealities because the activity coefficient of solid c o m p o n e n t i j ' lip equals u n i t y . Aselage et a l . (148) investigated the l i q u i d - s o l u t i o n behavior i n the w e l l - c h a r a c t e r i z e d G a - S b a n d I n - S b systems. T h e availab i l i t y of a t h e r m o d y n a m i c a l l y consistent data base (measurements of l i q u i d u s , c o m p o n e n t activity, a n d e n t h a l p y of mixing) p r o v i d e d the o p p o r t u n i t y to examine a variety of solution models. L i t t l e difference was f o u n d a m o n g seven models i n t h e i r ability to fit the c o m b i n e d data base, although a s y m m e t r i c models are expected to p e r f o r m better i n some systems. Often, a complete general, the results of p o n e n t activities from m o d e l parameters are

data base is unavailable for a p a r t i c u l a r system. I n a t t e m p t e d cross p r e d i c t i o n s (e.g., p r e d i c t i o n of c o m a fit to l i q u i d u s data) are unsatisfactory (190). W h e n a l l o w e d to vary w i t h t e m p e r a t u r e , a significant cor-

relation b e t w e e n the parameters is i n t r o d u c e d . A s an example, 8 7 % - c o n f i d e n c e ellipses for s i m p l e - s o l u t i o n parameters d e t e r m i n e d from fits to the c o m b i n e d G a - S b data set a n d l i q u i d u s alone are s h o w n i n F i g u r e 21. T h e i m p o r t a n c e of the ellipses is i n the shape rather than the relative m a g n i t u d e . T h e ellipse r e s u l t i n g from the l i q u i d u s fit i n F i g u r e 21 is q u i t e n a r r o w a n d elongated. T h i s e l l i p s e shape indicates a h i g h degree of correlation b e t w e e n the parameters. T h u s a w i d e range of values exists for the parameter a, a n d



3.

ANDERSON

Liquid-Phase

-15

Epitaxy

-9 -3 3 9 Percent change in a

153

15

Figure 21. Ellipses at 87% confidence for the simple-solution parameters a and b determined by a fit of the Ga-Sb data base: —, combined data set; —, liquidus data only. (Reproduced with permission from reference 190. Copy right 1985 Pergamon.) associated w i t h each value of α is a n a r r o w range for the p a r a m e t e r b. T h e s e parameters w o u l d give nearly the same d e s c r i p t i o n of the l i q u i d u s . A s l o n g as the p r o p e r t y to b e calculated requires the s u m a +

bT i n the m o d e l

equation (as i n the case of i s o t h e r m a l activity coefficient), this correlation is not a major p r o b l e m . T h i s result helps e x p l a i n the difficulty i n extracting reliable information o n the e n t h a l p y of m i x i n g from a fit to the l i q u i d u s ; solution models are imperfect i n t h e i r ability to represent the c o m p l e x so lution chemistry. Solid-Solution Models. C o m p a r e d w i t h the l i q u i d phase, v e r y few direct e x p e r i m e n t a l d e t e r m i n a t i o n s of the t h e r m o c h e m i c a l properties of c o m p o u n d - s e m i c o n d u c t o r solid solutions have b e e n r e p o r t e d . R a t h e r , procedures for c a l c u l a t i n g phase diagrams have r e l i e d o n two methods for estimating solid-solution m o d e l parameters. T h e first m e t h o d uses s e m i e m p i r i c a l relationships to describe the enthalpy of m i x i n g o n the basis of the k n o w n p h y s i c a l properties o f the b i n a r y c o m p o u n d s (202, 203). T h i s approach does not p r o v i d e an estimate for the excess e n t r o p y of m i x i n g a n d thus


154


neglects c l u s t e r i n g that has b e e n suggested i n some systems (204). T h e second m e t h o d estimates solution m o d e l parameters from a fit to e x p e r i m e n t a l values o f the m u l t i c o m p o n e n t s o l i d - l i q u i d phase d i a g r a m (142). T h i s p r o c e d u r e is subject to uncertainties i n t h e l i q u i d - s o l u t i o n t h e r m o d y n a m i c properties, t h e e x p e r i m e n t a l solidus a n d l i q u i d u s t e m p e r a t u r e values, a n d the appropriateness o f the solution m o d e l . Semiempirical Modek.

A t t e m p t s have b e e n made to calculate the s o l i d -

a n d l i q u i d - i n t e r a c t i o n parameters from t h e p h y s i c a l properties o f the c o n stituents. Ilegems a n d Pearson (163), F o s t e r (205), a n d P a n i s h et a l . (142) Downloaded by UNIV OF PITTSBURGH on May 14, 2016 | http://pubs.acs.org Publication Date: May 5, 1989 | doi: 10.1021/ba-1989-0221.ch003

have suggested that t h e t h e solid-phase i n t e r a c t i o n parameter c a n b e d e t e r m i n e d a p p r o x i m a t e l y from t h e m a g n i t u d e o f the m i s m a t c h b e t w e e n t h e lattice parameters o f the two e n d c o m p o u n d s , although n o analytical expressions w e r e p r e s e n t e d to quantify the c o n t r i b u t i o n to the excess G i b b s energy. F o r example,

t h e solid-interaction parameters

m i g h t b e zero for t h e

A l ^ G a ^ - V ternary systems, because t h e lattice parameters o f the b i n a r y c o m p o u n d s are n e a r l y i d e n t i c a l . U s i n g t h e P h i l l i p s - V a n V e c h t e n t h e o r y (206) o f c h e m i c a l b o n d i n g to calculate t h e solid-interaction parameters a n d t h e m o l a r v o l u m e s , H i l d e brand's s o l u b i l i t y parameters (207), a n d electronegativities (208, 209) o f the constituent elements to calculate t h e l i q u i d - i n t e r a c t i o n parameters, S t r i n g fellow (203, 210) calculated t h e b i n a r y a n d ternary phase diagrams o f group I I I - V systems. H o w e v e r , t h e agreement w i t h t h e e x p e r i m e n t a l l y d e t e r m i n e d phase boundaries was p o o r i n several cases. Stringfellow (203, 211) p r e s e n t e d a s i m p l e r , m o r e accurate s e m i e m p i r i c a l m o d e l , c a l l e d t h e d e l t a - l a t t i c e - p a r a m e t e r ( D L P ) m o d e l , w h i c h is based o n the P h i l l i p s t h e o r y (212) o f p r e d i c t i n g the solid-interaction parameters for group I I I - V systems. T h e results o f the calculations are i n good agreement w i t h those d e t e r m i n e d b y fitting t h e e x p e r i m e n t a l phase d i a g r a m . T h i s m o d e l , h o w e v e r , is n o t always appropriate for group I I - V I a n d other systems, because t h e D L P m o d e l neglects mismatches i n t h e ionicities a n d d e h y b r i d i z a t i o n factors o f the t w o b i n a r y c o m p o u n d s (212). F e d d e r s a n d M u l l e r (213) have d e r i v e d a n estimate o f the s o l i d - i n t e r action parameter from another p o i n t o f v i e w , w h i c h ascribes t h e m i x i n g e n t h a l p y to b o n d distortions associated w i t h t h e alloy formation a n d relates these distortions to t h e macroscopic elastic properties o f the crystal. T h e y c o n c l u d e d that the results based o n elastic-crystal parameters y i e l d a s i m i l a r form for t h e t h e r m o d y n a m i c properties as those estimated b y D L P m o d e l based o n optical-crystal parameters. T h e s e s e m i e m p i r i c a l models postulate that local strain associated w i t h different atomic sizes o f the elements is t h e major c o n t r i b u t i o n to t h e s o l i d solution e n t h a l p y o f m i x i n g . A n estimate o f lattice strain energy has b e e n c o m p a r e d to fitted values o f the enthalpy o f m i x i n g for several group I I I - V systems (156). T h e results l e d to a calculated enthalpy o f m i x i n g that was a


3.

ANDEHSON

Liquid-Phase

155

Epitaxy

factor o f 5 too h i g h . I n this elastic m o d e l (156), the strain e n e r g y was assumed to be that r e q u i r e d to increase the b o n d distance of one group I I I - V p a i r a n d decrease the b o n d distance of the other p a i r to achieve the average b o n d distance suggested b y the alloy lattice constant. E x t e n d e d X - r a y a b s o r p t i o n fine-structure

measurements of g r o u p I I I - V systems (214-216)

show that a

constant difference b e t w e e n b o n d lengths always exists at any g i v e n alloy c o m p o s i t i o n . A n expression for the strain e n e r g y was d e r i v e d that was c o n sistent w i t h the regular-solution f o r m u l a t i o n , the D L P m o d e l , a n d the fitted values of the e n t h a l p y of m i x i n g . F u r t h e r support for the D L P m o d e l a n d


estimates d e v e l o p e d from an assessment o f the phase d i a g r a m are f o u n d i n recent dissolution c a l o r i m e t r i c measurements of the e n t h a l p y of m i x i n g of G a ^ I n ^ P ( F i g u r e 22) (202, 203, 217-219)

H o w e v e r , some e v i d e n c e indicates

that the excess e n t r o p y of m i x i n g is not zero (217).

Figure 22. Enthalpy of mixing GaP(s) and InP(s) at 1048 K. The data were determined by dissolution calorimetry ( ; 217) and from references 202 ( ) 203 (—), 218 ( ), and 219 (—). (Reproduced with permission from reference 217. Copyright 1987 Ekevier.) ?


156



Parameter Estimation from Phase Diagram Assessment. T h e second approach u s e d to d e t e r m i n e solid-solution behavior is to estimate solution m o d e l parameters from a fit to the m e a s u r e d phase diagram. M a n y of the solution models u s e d to describe the l i q u i d solution have b e e n used to m o d e l the solid m i x t u r e . T h e s i m p l e - s o l u t i o n expression a n d its special cases have b e e n u s e d most extensively. T h e data base generally u s e d is e i t h e r the pseudobinary phase d i a g r a m or group I l l - r i c h portions of the ternary phase d i a g r a m . F o s t e r a n d coworkers (156, 220-222) r e p o r t e d that six p s e u d o b i n a r y sections can be satisfactorily fitted o n the assumptions that the l i q u i d phase is i d e a l a n d the solid phase is a t h e r m a l . P a n i s h a n d Ilegems (142) o b t a i n e d somewhat p o o r e r fits o n the a s s u m p t i o n that b o t h l i q u i d a n d s o l i d solutions are strictly regular. B r e b r i c k a n d P a n l e n e r (159) investigated the i d e a l , strictly regular, a t h e r m a l , a n d quasi regular models for each phase a n d c o n c l u d e d that the strictly regular l i q u i d w i t h the s i m p l e s o l i d is the simplest f o r m u l a t i o n g i v i n g satisfactory fits for each of seven systems e x a m i n e d . A general result o f p r e d i c t i n g m u l t i c o m p o n e n t phase diagrams from a fit to b i n a r y a n d p s e u d o b i n a r y or ternary phase diagrams is that l i q u i d u s isotherms are i n fair or good agreement w i t h e x p e r i m e n t a l data a n d are insensitive to the choice of m o d e l . T h e calculated solidus isotherms, h o w ever, often give fair agreement o n l y w i t h e x p e r i m e n t a l data a n d appear to be m o r e sensitive to the values of the interaction energies. A s an example, G r a t t o n a n d W o l l e y m e a s u r e d solidus isotherms i n the G a - I n - S b system (160). A l t h o u g h the parameter set chosen b y these authors, as w e l l as b y several others (142, 161, 162, 223), s h o w e d reasonable agreement w i t h exp e r i m e n t a l l i q u i d u s isotherms, the agreement w i t h the solidus isotherms was p o o r i n each case. O n e of the m a i n difficulties w i t h u s i n g the pseudobinary phase d i a g r a m as a data base for e s t i m a t i n g the solid solution properties is that the phase diagram represents the h i g h - t e m p e r a t u r e behavior only. F o r most a p p l i c a tions, the l o w e r t e m p e r a t u r e p o r t i o n of the phase diagram is i m p o r t a n t (e.g., for L P E a n d the p r e d i c t i o n of m i s c i b i l i t y gaps i n the s o l i d solution). T h e t e m p e r a t u r e d e p e n d e n c e of the solid-solution G i b b s excess energy is s e n sitive to the solution m o d e l a n d the m e t h o d of data r e d u c t i o n used. A s an example, C h a n g et a l . (224) s t u d i e d the G a ^ I n ^ S b p s e u d o b i n a r y system. T h e Ga .In ^ . S b l i q u i d m i x t u r e was treated e i t h e r as a ternary m i x t u r e of G a , I n , a n d S b , w i t h the t h e r m o d y n a m i c properties estimated w i t h b i n a r y parameters, o r as a b i n a r y m i x t u r e of G a S b a n d I n S b , w i t h the t h e r m o d y n a m i c properties calculated from the s i m p l e - s o l u t i o n m o d e l b y u s i n g the parameters estimated from a fit of the p s e u d o b i n a r y phase d i a g r a m . F o r b o t h descriptions of the l i q u i d m i x t u r e , the s i m p l e - s o l u t i o n equation was u s e d to m o d e l the solid-solution behavior, a n d parameters w e r e estimated from a fit of the p s e u d o b i n a r y phase diagram. B o t h treatments of the l i q u i d phase gave standard deviations i n the l i q u i d u s a n d solidus temperatures A


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w i t h i n the e x p e r i m e n t a l uncertainty. T h e variations of the G i b b s excess energy of the solid solution w i t h t e m p e r a t u r e , h o w e v e r , w e r e i n opposite directions for the two different treatments of the l i q u i d phase. T h u s the l o w t e m p e r a t u r e segment of the ternary phase diagram is p r e d i c t e d differently b y each assumption. Experimental Values of Γ ψ

A n e x p e r i m e n t a l value for Γ y can be as

signed b y solving the e q u i l i b r i u m equations for T

ir

I n the case of a t e r n a r y


e q u i l i b r i u m , the result is g i v e n b y

Γ

0

= ^

β

Χ

ρ ( θ

ΰ

)

(44)

B y u s i n g data from e x p e r i m e n t a l phase diagram to obtain values of χ

ίρ

x

it

a n d Xj a n d methods I - I V to calculate the q u a n t i t y θ y, values of Γ calculated for the A l - G a - S b a n d G a - I n - S b systems. Values of 1 -

Gasb w e r e

I GaSb are s h o w n as a function of c o m p o s i t i o n along several isotherms i n F i g u r e s 23 a n d 24. These two systems represent two extremes i n the solid-solution behavior, w i t h the A l ^ G a ^ S b system d e v i a t i n g o n l y slightly from i d e a l b e havior a n d the G a J n ^ S b system s h o w i n g strong positive deviations from Raoult's law. A s s h o w n i n F i g u r e 23, the value of T i n the A l - G a - S b system is significantly different from u n i t y as a result of nonidealities i n the l i q u i d solution. T h e value of r i n this system is also nearly i n d e p e n d e n t of b o t h c o m p o s i t i o n and t e m p e r a t u r e . T h u s this system can be m o d e l e d as an i d e a l G a S b

G a S b

solution i n b o t h phases i f the value of 6 was suitably adjusted. S h o w n i n F i g u r e 24 (160, 225) is a similar plot of 1 - r versus x along six isotherms for the G a - I n - S b system. T is seen to be a m o r e c o m p l e x function of b o t h t e m p e r a t u r e a n d c o m p o s i t i o n , a l t h o u g h this q u a n tity appears to be i n d e p e n d e n t of c o m p o s i t i o n along several isotherms. A closer inspection of equation 44 reveals that the quantity exp (θ^) is constant along an i s o t h e r m a n d that the quantities x^/Xi a n d 1/Xj are equal to twice the d i s t r i b u t i o n coefficients for elements i a n d j , respectively. T h u s , not s u r p r i s i n g l y , the variation of Ty w i t h c o m p o s i t i o n a n d t e m p e r a t u r e is not as p r o n o u n c e d as that of the i n d i v i d u a l activity coefficients. T h i s partial c a n cellation of the composition a n d t e m p e r a t u r e d e p e n d e n c e of the l i q u i d solution activity coefficient p r o d u c t b y the d e p e n d e n c e of the solid solution activity coefficient is largely responsible for the ability of rather s i m p l e so l u t i o n models to represent phase diagrams, a n d makes extraction of m e a n ingful estimates of the i n d i v i d u a l m i x t u r e properties f r o m fitted phase diagrams difficult. G a S b

G a S b

sh

G a S b

Binodal and Spinodal Decompositions.

A s the temperature of a

solid solution is l o w e r e d , the e n t r o p y c o n t r i b u t i o n to the s o l i d G i b b s energy


158



0.8

SX in ο

!

Figure 23. Experimental values of 1 - Tcasb as a function of xsb along several isotherms in the Al-Ga-Sb system. The calculation used the recommended value of Q b in Table I and the phase diagram measurements of Dedegkaev et al. (145) 778 Κ, Ο , 825 Κ, Δ , 873 Κ) and of Cheng and Pearson (182) ( Ο , 773 Κ; •, 823 Κ). GaS

is decreased. T h i s decrease i n G i b b s energy can p e r m i t m o r e - o r d e r e d phases to be stable a n d be i n s o l u b l e i n c o m p o u n d - s e m i c o n d u c t o r solid solutions. A t y p i c a l p s e u d o b i n a r y phase diagram e x h i b i t i n g a m i s c i b i l i t y gap is s h o w n i n F i g u r e 25. T w o different two-phase regions are d e p i c t e d . A t a h i g h t e m perature, the n o r m a l l i q u i d - s o l i d ( L + S) two-phase field is formed, b u t b e l o w a critical t e m p e r a t u r e , T , a two-solid field is formed. T h e solid-phase b o u n d a r y l i n e represents the m i s c i b i l i t y or b i n o d a l l i n e a n d is d e t e r m i n e d b y the c o n d i t i o n of e q u a l c h e m i c a l potentials i n each s o l i d phase. F o r regular solid-solution behavior, this l i n e is s y m m e t r i c w i t h respect to the axis χ = c



3.

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Epitaxy

.0 Ό

0.5 A

Sb

Figure 24. Experimental values of 1 - r s& as a function of x b along several isotherms in the Ga-In-Sb system. The calculation used the recommended values ofQ sb listed in Table I and the phase diagram measurements ofAntypas (225) ( O , 773 K) and of Gratton and Wolley (160) (•, 653 Κ; Δ , 703 K; O , 748 K; 873 K ; + , 923 K). S

Gfl

Ga

V2 and is represented by the equation

^(2x -

f_J_\ \1 -

xj

R

1)

2T (2x c

-1)

Τ

The spinodal line is determined by the state at which the second derivative of the Gibbs energy with respect to composition is equal to zero. For a



160


regular solution, the s p i n o d a l l i n e is d e t e r m i n e d b y

x(l

-

RT x) = — 2w

(46)

)4

T h e solutions i n the r e g i o n i n s i d e the spinodal d o m a i n are unstable, whereas the solid solutions i n the r e g i o n b e t w e e n the b i n o d a l a n d spinodal lines are metastable. T h e presence of a m i s c i b i l i t y gap l i m i t s the p o t e n t i a l usefulness of these materials i n device applications. Solutions w i t h compositions l y i n g inside the s p i n o d a l d o m a i n cannot be g r o w n b y L P E , whereas metastable solid solutions have a t e n d e n c y toward phase separation a n d , e v e n t u a l l y , device degradation. T h r e e types of b e h a v i o r are e x h i b i t e d b y c o m p o u n d - s e m i c o n d u c t o r sys tems (226). F i r s t , some systems do not i n v o l v e a m i s c i b i l i t y gap (e.g., A ^ G a ^ A s ) . S e c o n d , a class of p s e u d o b i n a r y phase diagrams show a m i s c i b i l i t y gap w i t h T b e l o w the solidus (e.g., G a J n ^ A s , G a J n ^ P , a n d G a P j A s ^ . ; F i g u r e 25). T h i r d , i n i m m i s c i b l e systems, the b i n o d a l c u r v e can penetrate the solidus l i n e to p r o d u c e a p e r i t e c t i c - t y p e phase diagram (e.g., Αΐ Ρ .8^_ . a n d G a P ^ S b j . J . S e v e r a l attempts have b e e n made to p r e d i c t b i n o d a l a n d s p i n o d a l curves for c o m p o u n d semiconductors from s e m i e m p i r i c a l models a n d analysis of the phase diagrams (227-231). B y u s i n g these methods, the values of T r e p o r t e d b y different authors for a particular system show considerable scatter w h e n fitting phase diagrams to d e t e r m i n e the solid-solution behavior. T h e value of T is sensitive to the m o d e l a n d data c

ν

Α

χ

c

n


3.

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Liquid-Phase

161

Epitaxy

base used. A n e x p e r i m e n t a l d e t e r m i n a t i o n of the b i n o d a l

decomposition

curves is difficult because of slow transformation kinetics i n the s o l i d phase.

Summary L i q u i d - p h a s e epitaxy is a m a t u r e process that is r e c e i v i n g intense c o m p e tition from alternative processes such as M B E a n d M O C V D . A l t h o u g h L P E is a small-scale operation, its a b i l i t y to p r o d u c e h i g h - p u r i t y , low-defectdensity films has m a d e this process a useful deposition t e c h n i q u e for m a n y


o p t o e l e c t r o n i c - d e v i c e applications. T h e future of L P E as a v i a b l e c o m m e r c i a l process is u n c e r t a i n a n d w i l l d e p e n d on the progress t o w a r d i m p r o v i n g the film q u a l i t y i n alternative processes. L P E is a m e t h o d that is r i c h i n c h e m i c a l e n g i n e e r i n g process p h e n o m e n a ,

i n c l u d i n g interfacial a n d transport p h e

n o m e n a a n d phase e q u i l i b r i a . M a n y of the c h e m i c a l processes are not f u l l y understood,

although

first-order

models a n d a vast amount of laboratory

experience have r e n d e r e d this m e t h o d reliable for the g r o w t h of a variety of alloys. A general f o r m u l a t i o n of the p r o b l e m of s o l i d - l i q u i d phase e q u i l i b r i u m i n quaternary systems was p r e s e n t e d a n d r e q u i r e d the evaluation of two t h e r m o d y n a m i c quantities, θ y a n d Ty. F o u r methods for calculating θ y f r o m e x p e r i m e n t a l data w e r e suggested. W i t h these methods, reliable values of θ y for most c o m p o u n d semiconductors

could be determined. The term Γ y

involves the d e v i a t i o n of the l i q u i d solution from i d e a l b e h a v i o r relative to that i n the solid. T h i s t e r m is less important than the i n d i v i d u a l activity coefficients because of a partial cancellation of the composition and t e m p e r ature d e p e n d e n c e of the i n d i v i d u a l activity coefficients. T h e t h e r m o d y n a m i c data base available for l i q u i d mixtures is far m o r e extensive than that for solid solutions. F u t u r e w o r k a i m e d at m e a s u r e m e n t of s o l i d - m i x t u r e p r o p erties w o u l d be h e l p f u l i n i d e n t i f y i n g m i s c i b i l i t y l i m i t s a n d t h e i r relation to L P E as a p r o b l e m of constrained e q u i l i b r i u m .

Abbreviations and Symbols

A

activity activities of c o m p o n e n t s i a n d j i n the s t o i c h i ometric liquid area

c

refers to a critical point w h e n u s e d as a s u b

C C C j, C i

script concentration e q u i l i b r i u m concentration concentrations of c o m p o n e n t i i n the l i q u i d a n d s o l i d , respectively

a #i , a / sl

1

e

1

S


162


C° C C

reference concentration of c o m p o n e n t i concentration i n the l i q u i d concentration i n the s o l i d

1

or C

s

s

C C C [i], C [j], C [ij]

i n i t i a l concentration constant-pressure m o l a r heat capacity constant-pressure m o l a r heat capacity of c o m -

àC

ponents i , j , a n d iy i n phase r difference b e t w e e n the m o l a r heat capacities of the p u r e l i q u i d s a n d the s o l i d c o m p o u n d

Q

p

p

r

p

r

p

r

p

AC [ij]

difference b e t w e e n the m o l a r heat capacities


p

of the stoichiometric l i q u i d a n d the p o u n d ij AC [n]

difference b e t w e e n the heat capacities of the

p

film

d d

2

p u r e l i q u i d and the s o l i d e l e m e n t η thickness e m p i r i c a l constants i n expression for

d, d

Xy

com-

3

molar

Dp Dj

heat capacity diffusion coefficients of components i a n d j , r e

e f

spectively refers to e q u i l i b r i u m w h e n u s e d as a subscript refers to formation w h e n u s e d as a subscript

G AG âG °(ij, Γ)

molar G i b b s energy molar G i b b s energy of formation standard m o l a r G i b b s energy of formation of

àG$

c o m p o u n d ij at t e m p e r a t u r e Τ difference b e t w e e n the m o l a r G i b b s energy of

G

the b u l k s o l i d a n d that of the l i q u i d solution excess i n t e g r a l m o l a r G i b b s energy of solution

f

f

x s

H AH

m i x

ΔΗ

η ι

molar enthalpy m o l a r e n t h a l p y of m i x i n g of stoichiometric l i q -

s l

uid mixture m o l a r e n t h a l p y of fusion of p u r e e l e m e n t η

η

ΔΗ(ί, χ/), A H ( j , i ) x

ij i, j , k / Ji k fci £ ff Κ e

relative partial molar e n t h a l p y of c o m p o n e n t i i n l i q u i d solution at c o m p o s i t i o n x* refers to s o l i d c o m p o u n d ij w h e n u s e d a s u p e r s c r i p t o r subscript refer to m i x t u r e components w h e n u s e d as s u perscripts or subscripts flux evaporation rate of c o m p o n e n t i first-order rate constant e q u i l i b r i u m d i s t r i b u t i o n coefficient, effective d i s t r i b u t i o n coefficient constant d e f i n e d b y equation 18

C^/Cj


1

3.

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Liquid-Phase

Epitaxy

163

constant w i t h value 0 i f e l e m e n t n is l i q u i d at Γ a n d 1 i f e l e m e n t η is s o l i d at Γ


1

refers to the l i q u i d w h e n u s e d as a superscript

I

l i q u i d height

m

refers to m e l t i n g w h e n used as a subscript

m

slope of l i q u i d u s

»»,orj M

slope of l i q u i d u s for c o m p o n e n t i or j molecular weight

M, mix

m o l e c u l a r w e i g h t of c o m p o n e n t i refers to a m i x i n g p r o p e r t y w h e n u s e d as a subscript

η Ν

n u m b e r of components i n l i q u i d phase rate of formation of n u c l e i refers to an i n i t i a l or reference value w h e n u s e d

0

as a subscript refers to the phase w h e n u s e d as a superscript

Ρ ρ,

partial d i s t r i b u t i o n coefficient of c o m p o n e n t ij

r r*

c o o l i n g rate critical nucleus radius

R s

gas constant refers to s o l i d w h e n u s e d as a superscript or

S

molar e n t r o p y standard m o l a r entropy of formation of s o l i d

subscript ÀS "[ij, r

TJ]

c o m p o u n d ij from the p u r e l i q u i d elements at the c o m p o u n d m e l t i n g t e m p e r a t u r e , T m

i j

standard m o l a r entropy of formation of s o l i d c o m p o u n d ij f r o m the elements i n t h e i r natu r a l state at the c o m p o u n d m e l t i n g t e m p e r si

ature, TJ refers to the stoichiometric l i q u i d w h e n u s e d as a superscript

t Γ

time temperature

J* r r T U m HP n m ΔΓ c

1

Λ

1

measurement temperature critical temperature liquidus temperature m e l t i n g t e m p e r a t u r e of c o m p o u n d ij m e l t i n g t e m p e r a t u r e of e l e m e n t η difference b e t w e e n saturation a n d actual t e m peratures

t? Φ) V

g r o w t h velocity velocity i n the y d i r e c t i o n volume


164


tv , w

interchange energy

Xi Xy

m o l e fraction of c o m p o n e n t i i n l i q u i d solution m o l e fraction of c o m p o n e n t ij i n s o l i d solution

xs

refers to an excess function w h e n u s e d as a

x, y

stoichiometry of s o l i d solution A ^ B ^ C ^ D ^

x

distance above g r o w t h interface

ζ δ

p r o p o r t i o n a l i t y constant (equation 1) d e v i a t i o n f r o m stoichiometry

ki

kj


superscript

7 7/, 7j

activity coefficient activity coefficients of components i a n d j i n the

!

7y Ty

liquid activity coefficient of c o m p o n e n t ij i n the s o l i d activity coefficient ratio (defined b y e q u a t i o n

θ

29) r e d u c e d standard-state c h e m i c a l p o t e n t i a l dif

s

υ

θΐ/> θsj , ©ij™, θ ^ 11

ference (defined b y e q u a t i o n 28) r e d u c e d standard-state c h e m i c a l p o t e n t i a l dif

ιν

ferences calculated b y methods I - I V μ

chemical potential

ΜΆ * μ h> M-c μ , μ , μ

Liquid-Phase Epitaxy and Phase Diagrams of Compound

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