Solid-Liquid Equilibrium in Ternary Group III-V Semiconductor Materials

14

Solid-Liquid Equilibrium in Ternary Group III-V Semiconductor Materials

1

T. L.Aselage ,K. M. Chang, and T. J. Anderson

Downloaded by CORNELL UNIV on May 18, 2017 | http://pubs.acs.org Publication Date: October 2, 1985 | doi: 10.1021/bk-1985-0290.ch014

College of Engineering and Chemical Engineering, University of Florida, Gainesville, FL 32611 A general approach to calculating s o l i d - l i q u i d phase diagrams of ternary Group III-V semiconductor mater i a l systems i s presented i n which the standard state chemical potential terms are evaluated separately from the l i q u i d and solid solution a c t i v i t y c o e f f i c i e n t terms. Three methods of determining a reduced standard state chemical potential change are given and their implementation i s illustrated for the Ga-Sb system. Knowledge of the deviation from ideal behavior of the l i q u i d mixture properties r e l a t i v e to the s o l i d solution i s important for calculating ternary phase diagrams and the c a l culation i s demonstrated for the Al-Ga-Sb system. The compounds formed by the Group I I I A elements o f the p e r i o d i c t a b l e , A l , Ga and I n , w i t h the group VA elements, P, As and Sb, have t h e p o t e n t i a l t o be e x t r e m e l y i m p o r t a n t semiconductor materials. The a t t r a c t i v e n e s s o f Group I I I - V compounds as e l e c t r o n i c m a t e r i a l s l i e s i n the v a r i a b i l i t y o f e l e c t r i c a l p r o p e r t i e s among the d i f f e r e n t compounds and the f a c t t h a t t h e s e p r o p e r t i e s are o f t e n s u p e r i o r t o those found i n S i . For d e v i c e s which r e q u i r e h i g h o p e r a t i n g speeds or f r e q u e n cies, several III-V materials offer a s i g n i f i c a n t l y larger elect r o n m o b i l i t y than S i . Indeed, f o r the h i g h m o b i l i t y compounds InAs and InSb the improvement i s g r e a t e r than an o r d e r o f magnitude. For o p t i c a l d e v i c e a p p l i c a t i o n s , i t i s advantageous t o utilize a m a t e r i a l w i t h a d i r e c t bandgap; the b i n a r y compounds GaAs, InP, GaSb, InAs and InSb have d i r e c t bandgaps, w h i l e S i has an i n d i r e c t bandgap. Because o f t h e more complex n a t i v e p o i n t d e f e c t s t r u c t u r e o f I I I - V m a t e r i a l s compared t o S i , most III-V compounds can be made s e m i - i n s u l a t i n g by c o n t r o l l e d p r o c e s s i n g .

1

Current address: Sandia National Laboratories, Albuquerque, N M 87185

0097-6156/85/0290-0276$06.00/0 © 1985 A m e r i c a n C h e m i c a l Society

Stroeve; Integrated Circuits: Chemical and Physical Processing ACS Symposium Series; American Chemical Society: Washington, DC, 1985.


14.

ASELAGE ET AL.

Ternary Group III-

V Semiconductor

Materials

277

In i n t e g r a t e d c i r c u i t s t h i s r e s u l t s i n lower power consumption and reduced p a r a s i t i c c a p a c i t a n c e s . Perhaps the most i m p o r t a n t p r o p e r t y o f I I I - V m a t e r i a l s , howe v e r , i s the a b i l i t y t o form c o m p l e t e l y m i s c i b l e s o l i d s o l u t i o n s f o r most o f the multicomponent systems. T h i s i s accomplished e i t h e r by the s u b s t i t u t i o n of an atom on the Group I I I s u b l a t t i c e , e.g. Ga, by another Group I I I element, e.g. I n , or by the substit u t i o n o f an atom on the Group V s u b l a t t i c e , e.g. As, by another Group V element, e.g. P. By the v a r i a t i o n o f the s o l i d solution c o m p o s i t i o n , the e l e c t r i c a l p r o p e r t i e s o f the s o l i d are a f f e c t e d . T h i s a f f o r d s the d e v i c e d e s i g n e r the a b i l i t y t o tune the electrical p r o p e r t i e s of the s o l i d t o f i t the p a r t i c u l a r d e v i c e r e q u i r e ments. In most a p p l i c a t i o n s , the d e s i g n e r w i l l specify the bandgap energy t o g i v e the d e s i r e d e l e c t r i c a l c h a r a c t e r i s t i c s and the l a t t i c e parameter t o p e r m i t the growth o f d e v i c e q u a l i t y epitaxial layers. To i n d e p e n d e n t l y vary both o f t h e s e m a t e r i a l parameters o f t e n r e q u i r e s a q u a t e r n a r y system. P r i m a r i l y because of s i m p l e r p r o c e s s i n g , c e r t a i n Group I I I - V t e r n a r y s o l i d composit i o n s have r e c e i v e d a t t e n t i o n . In some systems the c o m p o s i t i o n a l degree o f freedom i n the t e r n a r y s o l i d s o l u t i o n i s used t o match the l a t t i c e parameter t o t h a t o f the s u b s t r a t e ( e . g . Gp ^ y ^ g l a t t i c e matched t o I n P ) . In Ga A l ^ _ Y systems, However; the b i n a r y compounds have n e a r l y i d e n t i c a l l a t t i c e parameters. For these systems, the c o m p o s i t i o n a l degree o f freedom can then be used t o vary the bandgap energy, and t h e r e f o r e the e l e c t r i c a l p r o perties. Ternary S o l i d - L i q u i d

Equilibrium

Many o f the p r o c e s s i n g s t e p s i n the f a b r i c a t i o n o f d e v i c e s containing I I I - V m a t e r i a l s i n v o l v e the i n t e r f a c i a l contact of a l i q u i d phase w i t h a s o l i d phase. Examples o f such p r o c e s s e s a r e the melt or s o l u t i o n growth o f b u l k s i n g l e c r y s t a l s , the e p i t a x i a l growth o f t h i n f i l m s from s o l u t i o n , and melt purification p r o c e s s e s such as zone r e f i n i n g . As an e q u i l i b r i u m boundary cond i t i o n i s o f t e n i m p l i e d a t the s o l i d - l i q u i d interface, knowledge of the phase diagram f o r the m a t e r i a l system i s e s s e n t i a l i n the a n a l y s i s of these processes. In a d d i t i o n , the thermodynamic b e h a v i o r o f I I I - V s o l i d s o l u t i o n s i s i m p o r t a n t i n the a n a l y s i s o f p r o c e s s e s t h a t i n v o l v e a s o l i d - v a p o r i n t e r f a c e such as c h e m i c a l vapor d e p o s i t i o n . A d e s c r i p t i o n o f the s o l i d s o l u t i o n b e h a v i o r f o r I I I - V systems i s o f t e n o b t a i n e d from a r e d u c t i o n o f a v a i l a b l e s o l i d - l i q u i d equilibrium data. A g e n e r a l I I I - V s o l i d - l i q u i d t e r n a r y phase diagram i s depicted i n F i g u r e 1. The two I I I - V b i n a r y systems, A-C and B-C, show s i m i l a r b e h a v i o r . Each system forms an equimolar compound, AC or BC, which m e l t s a t a h i g h e r temperature than e i t h e r o f the pure elements (except f o r the InSb-Sb case). The b i n a r y phase diagram c o n s i s t s o f two s i m p l e e u t e c t i c systems on e i t h e r s i d e o f the compound ( e . g . , the A-AC and the AC-C s y s t e m s ) . The t h i r d b i n a r y phase diagram r e p r e s e n t s s o l i d - l i q u i d e q u i l i b r i u m between elements from the same group. In F i g u r e 1 the A-B p o r t i o n of the t e r n a r y phase diagram i s d e p i c t e d as b e i n g isomorphous


278

CHEMICAL AND PHYSICAL PROCESSING OF INTEGRATED CIRCUITS

AxBj_ C Pseudo-Binary Liquidus x


I

AxBi_ C Pseudo-Binary Solidus x

A

F i g u r e 1. Group I I I - V s o l i d - l i q u i d t e r n a r y phase

diagram.


14.

ASELAGE ET AL.

Ternary Group III-

V Semiconductor

Materials

279

though o t h e r s t r u c t u r e s e x i s t . The plane o f e q u a l number o f atoms from the Group I I I and V columns c o n t a i n s the l i n e compounds AC and BC and r e p r e s e n t s the A^B-^ C pseudobinary phase diagram. The solid-liquid e q u i l i b r i u m r e g i o n of the phase diagram i s isomorphous f o r a l l but one of the p o s s i b l e 18 III-V ternary systems. In systems w i t h a l a r g e d i f f e r e n c e .in atomic s i z e s , m i s c i b i l i t y gaps i n the s o l i d s o l u t i o n occur (not d e p i c t e d i n F i g u r e 1 ) . The o t h e r prominent f e a t u r e o f the t e r n a r y phase diagram shown i n F i g ure 1 i s the e u t e c t i c v a l l e y formed at c o m p o s i t i o n s r i c h i n C. The two-phase f i e l d , A C p l u s l i q u i d , i s the most i m p o r t a n t segment o f the phase diagram r*or p r o c e s s i n g c o n s i d e r a t i o n s . The s o l i d - l i q u i d e q u i l i b r i u m s t a t e o f the A-B-C t e r n a r y s y s tem i s c a l c u l a t e d by e q u a t i n g the temperature and p r e s s u r e o f each phase as w e l l as the c h e m i c a l p o t e n t i a l s o f each o f the s p e c i e s p r e s e n t i n both phases. In a d d i t i o n t o these e q u a t i o n s , a cons t r a i n t o f s t o i c h i o m e t r y i s p l a c e d on the s o l i d s o l u t i o n ; the sum of the mole f r a c t i o n s o f the Group I I I elements must be e q u a l t o the sum o f the mole f r a c t i o n s o f the Group V elements. Because o f this c o n s t r a i n t , the c h e m i c a l p o t e n t i a l s o f the t h r e e s p e c i e s are not i n d e p e n d e n t l y v a r i a b l e i n the s o l i d . The t e r n a r y s o l i d solut i o n A^B^ C can be t r e a t e d as i f i t were a b i n a r y s o l u t i o n o f components ^C and BC. The requirement o f equal c h e m i c a l potent i a l s o f each o f the s p e c i e s p r e s e n t i n both phases then becomes


x

UlC = Uj + u£

I = A,B

(1)

where I and C r e f e r t o the l i q u i d phase components, and IC refers to the s o l i d phase component. By imposing the c o n s t r a i n t of s t o i c h i o m e t r y on the s o l i d , the e f f e c t o f the p o i n t d e f e c t s t r u c t u r e i s n e g l e c t e d . The b i n a r y I I I - V compounds are not s t r i c t l y line compounds; some nons t o i c h i o m e t r y i s p r e s e n t . The c a l c u l a t i o n s o f H u r l e (1_) and Edel i n and M a t h i o t (_2) i n d i c a t e t h a ^ the r e g i o n o f n o n s t o i c h i o m e t r y i s q u i t e s m a l l ( l e s s than 10" mole f r a c t i o n ) i n GaAs and GaSb. B r e b r i c k O ) has d i s c u s s e d the v a r i a t i o n o f the Gibbs energy of the s o l i d f o r such n e a r l y s t o i c h i o m e t r i c compounds, d e m o n s t r a t i n g t h a t t h i s v a r i a t i o n i s n e g l i g i b l e f o r compounds w i t h homogeneity gaps o f l e s s than 1% i n mole f r a c t i o n . Thus, a l t h o u g h the chemic a l p o t e n t i a l of a component i n the s o l i d varies considerably between the l i m i t s o f s t o i c h i o m e t r y , the sum o f the element chemic a l p o t e n t i a l s i n the b i n a r y s o l i d i s n e a r l y c o n s t a n t . T h i s cons t a n t v a l u e f o r the sum o f the s o l i d element c h e m i c a l p o t e n t i a l s i s the term i n E q u a t i o n 1. The c h e m i c a l p o t e n t i a l o f component i i n phase p,u?, can be O

r e l a t e d to a standard chemical p o t e n t i a l ,

D

^~

, i n terms o f the

a c t i v i t y c o e f f i c i e n t , yP, and mole f r a c t i o n , xP, a c c o r d i n g t o P = °'P y

+ RT In y P x

p

(2)

The i n s e r t i o n o f E q u a t i o n 2 f o r each term i n E q u a t i o n 1, f o l l o w e d by a l g e b r a i c m a n i p u l a t i o n , r e s u l t s i n the f o l l o w i n g i m p l i c i t


C H E M I C A L A N D PHYSICAL PROCESSING OF INTEGRATED

280

CIRCUITS

e x p r e s s i o n s f o r t h e l i q u i d and s o l i d e q u i l i b r i u m mole f r a c t i o n s o f s p e c i e s A and AC J- . (i_ X

A

= r

A

exp(-9

C

X c

A C

exp(-9

) )

- r

B

C

B C

)

exp(-9

B C

(3)

)

and

X

X

AC

C

( 1

X

- C

W^P^BC

"

= "I

5

( 4 )

—1 e X p (


}

°AC

)

' 1^

^P^BC*

types o f v a r i a b l e s appear i n E q u a t i o n s 3 and 4, and T^. The v a r i a b l e 9 ^ i s a reduced s t a n d a r d s t a t e c h e m i c a l p o t e n t i a l d i f f e r e n c e and i s d e f i n e d by TVAIO

o,s Q

IC'-—

0,1

0,1

*T

—

I

=

A

'

B

The term V i s t h e r a t i o o f l i q u i d phase t o s o l i d phase c o e f f i c i e n t s and i s g i v e n by

r ic r

T P

z

y = -4-^ ~7 IC

( 5 )

activity

I = A,B

(6)

The problem o f d e s c r i b i n g the ternary I I I - V phase diagram i s reduced t o s e l e c t i n g t h e s t a n d a r d s t a t e s , d e t e r m i n i n g t h e temperat u r e and p r e s s u r e ( n e g l i g i b l e ) dependence o f 9 ^ and d e t e r m i n i n g the temperature, p r e s s u r e ( n e g l i g i b l e ) and c o m p o s i t i o n dependence of r . i c

Reduced Standard S t a t e Chemical P o t e n t i a l Change With t h e s t a n d a r d s t a t e f o r each component chosen as t h e pure component i n t h e phase o f i n t e r e s t and a t t h e temperature o f i n t e r e s t , Chang e t a l . (40 have d i s c u s s e d t h r e e thermodynamic sequences f o r t h e c a l c u l a t i o n o f the reduced s t a n d a r d s t a t e chemical potentials. The pathways f o r each sequence a r e shown i n F i g ure 2. In method I ( F i g u r e 2a) t h e temperature o f t h e s o l i d compound IC i s r a i s e d from t h e l i q u i d u s temperature, T, t o i t s m e l t i n g temp e r a t u r e , T . The s o l i d i s then m e l t e d , f o r m i n g a stoichiometric l i q u i d , which i s c o o l e d t o t h e l i q u i d u s temperature. F i n a l l y , t h e s t o i c h i o m e t r i c l i q u i d i s s e p a r a t e d i n t o t h e pure l i q u i d elements. The dashed l i n e i n F i g u r e 2a i n d i c a t e s t h a t t h e s e l a s t two s t e p s i n v o l v e a s t o i c h i o m e t r i c l i q u i d below i t s normal f r e e z i n g point, i.e. a subcooled l i q u i d . The sum o f t h e Gibbs energy changes f o r


Ternary Group HI- V Semiconductor

ASELAGE ET AL.

Materials

Calculation of B | Q

Sequence IC Melt


I I | Solid IC

T

m

Seperate

Stoichiometric Liquid

Liquid Elements

Melt

lorC

solid I liquid

Separate

Liquid Elements

Elements

Solid IC

Separate

Melt

Solid IC Stoichiometric Liquid

Liquid Elements

F i g u r e 2. T h r e e t h e r m o d y n a m i c s e q u e n c e s f o r e v a l u a t i n g t h e reduced standard state chemical potential change.


282


each of the steps i n the process provides the standard state chemi c a l potential difference, which i s given by

T 9

AS(IC) m R

In a ! ( T ) a ^ ( T ) 1

IC

m

-, ] "

1 1

+

m

T m

1

AC (IC) dT

RT

T

T

(7)

si, i n a stoichiometric Here a° (T) i s the a c t i v i t y of component i l i q u i d at the liquidus temperature, T, AS (IC) i s the entropy of fusion of compound IC at the melting temperature, T , and AC (IC) i s the difference i n heat capacity between t h e s t o i c h i o m i t r i c l i q u i d and the s o l i d compound. This sequence i s the same as that proposed for binary III-V systems by Vieland (_5). In method II, (Figure 2b), the s o l i d compound IC i s separated into the pure elements at the liquidus temperature, T. The Gibbs energy change for this step i s simply the Gibbs energy of formation of the compound at the liquidus temperature. If the pure elements are l i q u i d s at t h i s temperature, i t i s also proportional to the standard state chemical potential change. If one or both of the elements i s a s o l i d , a process of raising the temperature to the melting one, melting, and cooling to the liquidus temperature must be carried out. In this case, however, the sequence i s performed on the pure element rather than the stoichiometric mixture. The dashed l i n e i n Figure 2b indicates that the f i n a l step in the process involves subcooling one or both of the pure l i q u i d s from their melting temperatures to the liquidus temperature. With t h i s sequence the expression for the standard state chemical potential difference i s


m

r AG°(T) IC

RT

2

K. l RT

T '

m

m

A S ( i ) ( T ^ - T) m m m

AC^i) T

T

dT'

(8)

T

where AG^(T) i s the Gibbs energy of formation of the compound at the liquidus temperature, T, AS ( i ) i s the entropy of fusion of pure cpmoonent i , T i s the melting temperature of pure component i, Ac'*' i s the difference i n heat capacity between pure l i q u i d i and s o P i d i , and K. i s zero i f i i s a l i q u i d at the temperature T and one i f i i s a s o l i d at T. In method I I I , (Figure 2c), the temperature of s o l i d IC i s raised from the liquidus temperature to the melting temperature. The s o l i d i s then melted, forming the stoichiometric l i q u i d , which i s then separated into the pure l i q u i d elements. F i n a l l y , the pure l i q u i d components are cooled from the melting temperature of the compound to the liquidus temperature. If the melting temperatures of the pure elements are above the liquidus temperature, t h i s l a s t step again involves subcooling the pure elements. The equation for the standard state chemical potential difference resulting from the sequence i n Figure 2c i s


14.

Ternary Group

ASELAGE ET AL.

T

283

Materials

AS°(I)

T

m

r

T

"

1

AC°(IC)

_1_ RT

(9)

dT

T


V Semiconductor

m , sl/ \ sl, x -y- I n a ( T ) a (T ) I I m L m T

IC

HI-

T

si, i n the s t o i c h i o m e t r i c where a. (T ) i s the a c t i v i t y of component l i q u i d a t tfie compound m e l t i n g temperature AS^(IC) i s the e n t r o p y of f o r m a t i o n of compound IC from the pure l i q u i d s a t t h e i r m e l t i n g temperature, and AC (IC) i s the d i f f e r e n c e i n the heat c a p a c i t i e s of the pure l i q u i d s land the s o l i d compound. Each of the above t h r e e methods employs a d i f f e r e n t data base. Most of the p r o p e r t y v a l u e s r e q u i r e d f o r the e v a l u a t i o n of i n E q u a t i o n s 7-9 have been e x p e r i m e n t a l l y determined f o r I I I - V systems and these t h r e e r e l a t i o n s h i p s can be used as a t e s t f o r thermodynamic c o n s i s t e n c y . The f i r s t method, E q u a t i o n 7, i s most r e l i a b l e a t or near the b i n a r y compound m e l t i n g temperature. As the temperature i s lowered below the m e l t i n g one, u n c e r t a i n t i e s i n the e x t r a p o l a t e d s t o i c h i o m e t r i c l i q u i d heat c a p a c i t y and component a c t i v i t y c o e f f i c i e n t s become i m p o r t a n t . The second method, Equat i o n 8, i s l i m i t e d t o the temperature range i n which an e x p e r i mental d e t e r m i n a t i o n o f AG^ i s f e a s i b l e (e.g., high temperature galvanic c e l l ) . Method I I i s a l s o v a l u a b l e f o r " p i n n i n g down" the low temperature v a l u e s o f 9 j p . Method I I I i s the p r e f e r r e d p r o cedure when e s t i m a t i n g s o l u t i o n model parameters from l i q u i d u s d a t a . S i n c e the a c t i v i t y c o e f f i c i e n t s o f the s t o c h i o m e t r i c l i q u i d components a r e e v a l u a t e d o n l y at the m e l t i n g temperature, the temp e r a t u r e dependence o f 0JQ i s e x p l i c i t i n t h i s e x p r e s s i o n . An Example: Ga-Sb. As an example, the reduced standard s t a t e c h e m i c a l p o t e n t i a l was e v a l u a t e d by each method f o r the Ga-Sb s y s tem. The m e l t i n g temperature o f GaSb i s low (985K) r e l a t i v e to other I I I - V systems and the vapor p r e s s u r e a t t h i s temperature i s s m a l l when compared t o the a r s e n i d e s and phosphides. Because o f the e x p e r i m e n t a l a c c e s s i b i l i t y , v a l u e s f o r each o f the thermochemi c a l p r o p e r t i e s r e q u i r e d t o e v a l u a t e S Q ^ U w i t h E q u a t i o n s 7-9 have been measured a t v a r i o u s temperatures and o f t e n by s e v e r a l i n v e s tigators. The reduced s t a n d a r d s t a t e c h e m i c a l p o t e n t i a l f o r GaSb was c a l c u l a t e d by E q u a t i o n s 7 and 9 w i t h the e x p e r i m e n t a l p r o p e r t y v a l u e s s e l e c t e d t o g i v e maximum and minimum v a l u e s f o r ^Q sh* These upper and lower e x p e r i m e n t a l bounds are shown i n F i g u r e The thermochemical p r o p e r t y v a l u e s s e l e c t e d t o g i v e these bounds are summarized i n Table I . a

The d o t t e d l i n e i n F i g u r e 3 shows the v a l u e o f G calculated u s i n g E q u a t i o n 8 and the measurements o f G^(GaSD; By Abbasov e t al. ( 1 9 ) . The v a l u e of 8 a t 298K shown i n F i g u r e 3 (circle) was also calculated using Equation 8 w i t h the v a l u e of AG°(GaSb,298 K) taken from L i c h t e r and Sommelet ( 2 0 ) . A f o u r t h method by which G^p can be determined from e x p e r i In the mental r e s u l t s i s an a p p l i c a t i co n o f E q u a t i o n s 3 and 4. binary l i m i t ( X ^ solved 1, x 1 - x^) these e q u a t i o n s can be A f o r 9j£ t o g i v e p

G a S b


284


800 700

T(K) 500

600


Tm

400

298

4

I0 /T(K)

F i g u r e 3 . V a l u e s o f 0Q 5 v e r s u s r e c i p r o c a l t e m p e r a t u r e . , upper and lower b o u n d , E q u a t i o n 7: , E q u a t i o n 8, AGS(GaSb) f r o m r e f . ( 1 9 ) ; 0, E q u a t i o n 7, A G ° ( G a S b ) from r e f . ( 2 0 ) ; , u p p e r a n d l o w e r b o u n d , E q u a t i o n 9; , recommended v a l u e s o f 0Qoc b ; E q u a t i o n 10, a f r o m r e f . (15), T - x ' f r o m r e f . (21_), enthalpy o f mixing from r e f . ( 2 2 ) . a

b

G

a


14.

Ternary Group

ASELAGE ET AL.

Table I .

Materials

Summary o f Ga-Sb Thermochemical P r o p e r t y Used i n E q u a t i o n s 7 and 9

285

Values

Property

Upper Bound

Ref.

Lower Bound

Ref.

a^ (1003K)aei(1003K) La bb T (GaSb) K m AH (GaSb) kJ/mole

0.134

6

0.0293

14

975

7

998

15

8

66.94

16

1

50.21

m

-854

9

-1071

17

5.699

10

2.895

16

-23.24

11.12

-28.16

10,13

3.24-9.29 x 10 T

12.13

33.35-21.68 log T

13,18

s l


III- V Semiconductor

AH J/mole mix AC (GaSb) J/mole K p AS°(GaSb,T ) J/mole K m

0

AC (GaSb) J/mole K P

9

I C

1 1 ^ 1 ^ 1 = l n [ a ( T , x ) a ( T , x ) ] = l n t a ^ T , X j ) a ( T ,Xj) I

J

L

I

ATT (x\) — —

c

I

c

+ AH (xJ) dT RT r

L

?

1

(10)

Here, a,.(T, x^ )a^(T, x^) i s the a c t i v i t y product at the l i a u i d u s temperature, T, or a j some measurement temperature, T , and l i q u i d u s c o m p o s i t i o n , x^, and AH^(x^) i s the r e l a t i v e p a r t i a l molar e n t h a l p y f o r component i = I , C a t the l i q u i d u s c o m p o s i t i o n . The d a t a base r e q u i r e d f o r t h i s f i n a l procedure i s the phase diagram and the a c t i v i t y p r o d u c t a t the l i q u i d u s temperature and c o m p o s i t i o n (or an i s o t h e r m a l a c t i v i t y product and the l i q u i d phase e n t h a l p y o f m i x i n g ) . T h i s method r e q u i r e s b i n a r y m i x t u r e i n f o r m a t i o n and w i l l g i v e two v a l u e s o f 9 ^ a t each l i q u i d u s temp e r a t u r e , one from each s i d e o f the compound, which a r e i d e n t i c a l g i v e n a c o n s i s t e n t d a t a s e t . T h i s procedure was a p p l i e d t o f o u r measurements o f a^ (1003K) o f Anderson e t a l . (_6) a l o n g w i t h the i n t e r p o l a t e d phase cliagram o f Maglione and P o t i e r (21J and the e n t h a l p y o f m i x i n g r e s u l t s o f Gambino and Bros (22^). The r e s u l t s are p l o t t e d i n F i g u r e 3 w i t h square symbols.


286


As d i s p l a y e d i n F i g u r e 3, t h e r e e x i s t s a l a r g e v a r i a t i o n i n the v a l u e s which can be a s s i g n e d t o 9^ An e x a m i n a t i o n o f the r e s u l t s from a l l f o u r p r o c e d u r e s , though , suggests an a p p r o p r i a t e temperature dependence o f ^Q 5h' ^ °lid l i n e i n F i g u r e 3 r e p r e s e n t s the v a l u e s o f B ^ c ^ s e l e c t e d f o r t h i s s t u d y . 1

n e

s


a

D e t e r m i n a t i o n o f S t o i c h i o m e t r i c L i q u i d Component A c t i v i t i e s . The c a l c u l a t i o n o f the reduced standard state chemical p o t e n t i a l d i f f e r e n c e by E q u a t i o n 7 or 9 i n the absence o f knowledge o f the thermodynamic p r o p e r t i e s o f the l i q u i d phase n e c e s s i t a t e s the i m p o s i t i o n o f a s o l u t i o n model t o r e p r e s e n t the product o f the component a c t i v i t i e s i n the s t o i c h i o m e t r i c l i q u i d . In E q u a t i o n 7 the a c t i v i t y product i s r e q u i r e d as a f u n c t i o n o f temperature, w h i l e E q u a t i o n 9 r e q u i r e s t h a t i t be c a l c u l a t e d o n l y a t the m e l t i n g temperature o f the b i n a r y compound. When E q u a t i o n 7 or 9 i s used i n the c a l c u l a t i o n o f the phase diagram, a s o l u t i o n model i s a l s o r e q u i r e d t o determine the TTQ term i n E q u a t i o n s 3 and 4. R e p r e s e n t i n g 9^p by E q u a t i o n / and d e t e r m i n i n g the a c t i v i t y p r o duct w i t h a s o l u t i o n model has been the procedure used almost e x c l u s i v e l y i n the l i t e r a t u r e t o i n t e r p o l a t e , e x t r a p o l a t e and p r e d i c t I I I - V phase diagrams ( 2 3 ) . In o r d e r t o t e s t the e f f e c t o f u s i n g a s o l u t i o n model t o c a l c u l a t e v a l u e s o f 9 ^ , the Non-Random Two-Liquid (NRTL) e q u a t i o n (24) was used i n c o n j u n c t i o n w i t h E q u a t i o n 7 or 9 t o f i t Ga-Sb d a t a s e t s c o n s i s t i n g o f the l i q u i d u s temperature a l o n e , l i q u i d u s temperature and e n t h a l p y o f m i x i n g , l i q u i d u s temperature and i s o t h e r m a l Ga a c t i v i t y , and a l l t h r e e t y p e s o f d a t a combined. Using the parameters determined from the d a t a r e d u c t i o n , v a l u e s o f ^GaSb f tJ°f r e c i p r o c a l temperature were c a l c u l a t e d and compared w i t h the recommended v a l u e s g i v e n i n F i g u r e 3. The measured v a l u e s o f l i q u i d u s temperature, a c t i v i t y o f Ga, and e n t h a l p y of m i x i n g used i n the f i t are those s e l e c t e d by Aselage e t a l . (25) ; these d a t a s e t s have been shown t o be thermodynamically cons i s t e n t by Anderson e t a l . (6_). For the f i r s t case, E q u a t i o n 7 was used i n which the a c t i v i t y product o f the s t o i c h i o m e t r i c l i q u i d a t the l i q u i d u s temperature of i n t e r e s t was c a l c u l a t e d from the NRTL e q u a t i o n . The enthalpy of f u s i o n and m e l t i n g temperature f o r the compound as w e l l as the heat c a p a c i t y d i f f e r e n c e , however, were s p e c i f i e d . The values chosen f o r t h e s e p r o p e r t i e s were those recommended by Chang e t a l . (4_). F i g u r e 4 compares the c a l c u l a t e d v a l u e s o f ^Qgc^ f° each d a t a s e t w i t h the recommended v a l u e s ( s o l i d l i n e ) . I n these d a t a r e d u c t i o n s the NRTL e q u a t i o n had f o u r a d j u s t a b l e parameters and the non-randomness f a c t o r was f i x e d a t -0.001. The r e s u l t s g i v e n i n F i g u r e 4 are an extreme example o f the p o s s i b l e d i s c r e p a n c y between the v a l u e s o f 9 ^ c a l c u l a t e d by t h i s procedure and the recommended v a l u e s . The a b i l i t y o f the parameter e s t i m a t e s o b t a i n e d i n these f i t s t o reproduce the d a t a s e t s i s w i t h i n the r e p o r t e d e x p e r i m e n t a l e r r o r f o r each data s e t . Thus, e r r o r s i n the e x t r a p o l a t e d v a l u e s o f 9^ are c a n c e l e d by e r r o r s i n the c a l c u l a t e d v a l u e s o f r , ^ . U s i n g v a l u e s o f 9 ^ determined i n t h i s way to e x t r a p o l a t e d a t a s e t s i n temperature or t o p r e d i c t multicoma s

3

u n c

o n

r


14.

Ternary Group

ASELAGE ET AL.

III- V Semiconductor


GaSb

Materials

T/K

m900

700

1.0

1.4

1.2

600 550500

1.6

1.8

2.0

450

2.2

400

2.4

F i g u r e 4. V a l u e s o f 6Q a c b v e r s u s r e c i p r o c a l t e m p e r a t u r e a s c a l c u l a t e d b y t h e N R T L e q u a t i o n a n d E q u a t i o n 7. R e d u c t i o n o f d a t a set: , liquidus temperature; , l i q u i d u s temperat u r e and Ga a c t i v i t y ; , l i q u i d u s temperature and e n t h a l p y of mixing; , l i q u i d u s temperature, e n t h a l p y o f mixing and Ga a c t i v i t y ; , r e c o m m e n d e d v a l u e s o f ©jQasb-


287

288


ponent phase diagrams and s o l u t i o n b e h a v i o r would g i v e u n r e l i a b l e results. For the second case, E q u a t i o n 9 was used w i t h E q u a t i o n s 3 and 4 t o reduce the same b i n a r y Ga-Sb d a t a s e t s . The s t o i c h i o m e t r i c l i q u i d m i x t u r e a c t i v i t y product was f i x e d a t the m e l t i n g temperat u r e by the NRTL model parameters. The r e m a i n i n g thermodynamic p r o p e r t i e s found i n E q u a t i o n 9 were a s s i g n e d the v a l u e s recommended by Chang e t a l . (4.). The r e s u l t s o f these c a l c u l a t i o n s are p r e s e n t e d i n F i g u r e 5. The temperature dependence o f 8 r 5 c a l c u l a t e d w i t h any o f the d a t a s e t s i s the same as t h a t f o r the s e l e c t e d v a l u e s o f ^gc^* T h i s i s a r e s u l t o f the f a c t t h a t E q u a t i o n 9 r e q u i r e s the c a l c u l a t i o n o f the a c t i v i t y p r o d u c t o n l y a t the compound m e l t i n g temperature. A r e a s o n a b l y c l o s e p r e d i c t i o n o f the a c t i v i t y product by the s o l u t i o n model s h o u l d g i v e a good r e p r e s e n t a t i o n o f the temp e r a t u r e dependence. E q u a t i o n s 3 and 4 f o r a b i n a r y system a t the m e l t i n g temperature reduce t o


a

9

T P

1L

= In

1

b

( T j a ^ U ) m L m

(11)

S i n c e the m e l t i n g temperature i s i n c l u d e d i n the d a t a s e t f o r each case s t u d i e d , a c l o s e p r e d i c t i o n o f the a c t i v i t y product a t the m e l t i n g temperature i s e x p e c t e d , p a r t i c u l a r l y i f component a c t i v i t i e s are a l s o i n c l u d e d i n the d a t a s e t . Determination of E q u i l i b r i u m between a pseudobinary I I I - V s o l i d s o l u t i o n and a t e r nary l i q u i d s o l u t i o n i s d e s c r i b e d by E q u a t i o n s 3 and 4. By the methods p r e s e n t e d i n the p r e v i o u s s e c t i o n , the d e t e r m i n a t i o n o f the reduced s t a n d a r d s t a t e c h e m i c a l p o t e n t i a l change, 9 ^ , can proceed i n a r e l i a b l e manner. The o t h e r term c o n t a i n e d i n Equat i o n s 3 and 4 i s T ^ , and i t s d e t e r m i n a t i o n i s d i s c u s s e d here. L i q u i d S o l u t i o n B e h a v i o r . The component a c t i v i t y c o e f f i c i e n t s i n the l i q u i d phase can be addressed s e p a r a t e l y from those i n the s o l i d s o l u t i o n by d i r e c t 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 or by a n a l y s i s of the b i n a r y l i m i t s , s i n c e )^ = 1. Because o f the l a r g e amount of e x p e r i m e n t a l e f f o r t r e q u i r e d t o study a t e r n a r y c o m p o s i t i o n f i e l d and the h i g h vapor p r e s s u r e s encountered i n the a r s e n i d e and phosphide m e l t s , a d i r e c t 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 o f t e r n a r y a c t i v i t y c o e f f i c i e n t s has been r e p o r t e d o n l y f o r the Ga-In-Sb s y s tem (26). T y p i c a l l y , the a v a i l a b l e b i n a r y l i q u i d u s d a t a have been used t o f i x the a d j u s t a b l e parameters i n a s o l u t i o n model w i t h 9 , ^ determined by E q u a t i o n 7. The s o l u t i o n model e x p r e s s i o n f o r the activity c o e f f i c i e n t has been used not o n l y t o r e p r e s e n t the component a c t i v i t i e s a l o n g the l i q u i d u s c u r v e , but also the s t o i c h i o m e t r i c l i q u i d a c t i v i t i e s needed i n E q u a t i o n 7. The t e r nary melt s o l u t i o n b e h a v i o r i s then o b t a i n e d by e x t e n d i n g the b i n a r y models t o d e s c r i b e a t e r n a r y m i x t u r e w i t h o u t a d d i t i o n a l a d j u s t a b l e parameters. In g e n e r a l , i n t e r a c t i o n s between atoms i n d i f f e r e n t groups e x h i b i t n e g a t i v e d e v i a t i o n s from i d e a l b e h a v i o r


14.

ASELAGE ET AL.

Ternary Group HI- V Semiconductor

Materials

289

w h i l e i n t e r a c t i o n s between atoms i n the same group show p o s i t i v e deviations. A number o f attempts have been made t o use the s i m p l e solut i o n model t o r e p r e s e n t the s o l u t i o n n o n i d e a l i t i e s i n b i n a r y (23,29-32) and t e r n a r y (23,33-41) I I I - V systems. I n the simple s o l u t i o n model, the i n t e g r a l Gibbs excess energy i s g i v e n i n terms of an i n t e r a c t i o n energy co(T) by the e q u a t i o n


G

X S

= o)(T) x ( l - x ) A

A

(12)

where o)(T) i s u s u a l l y g i v e n a l i n e a r temperature dependence, a+bT. The b i n a r y phase diagrams have been f i t t e d w e l l u s i n g this model, a l t h o u g h d i f f e r e n t v a l u e s o f the i n t e r a c t i o n parameter were n e c e s s a r y f o r the c o m p o s i t i o n range on e i t h e r s i d e o f the compound for the h i g h l y asymmetric Al-Sb system ( 3 4 ) . In s p i t e o f a wide v a r i e t y o f parameters r e p o r t e d f o r many o f the b i n a r y systems, the liquid phase thermodynamic p r o p e r t i e s d e r i v e d from any o f the parameter s e t s are i n v a r i a n c e w i t h a v a i l a b l e measurements of these p r o p e r t i e s . For example, 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 from a f i t of the l i q u i d u s i s always p o s i t i v e , whereas the availa b l e e x p e r i m e n t a l d a t a a l l show n e g a t i v e v a l u e s . I t i s a n t i c i pated t h a t the m i x i n g e n t h a l p y i s indeed n e g a t i v e i n a l l I I I - V b i n a r y systems, due t o the a t t r a c t i v e n a t u r e o f the I I I - V i n t e r a c tion. Similarly, the v a l u e s o f the i n t e r a c t i o n coefficients derived from the l i q u i d u s f i t s are i n poor agreement w i t h those d e r i v e d from vapor p r e s s u r e measurements i n the a r s e n i d e and phosphide systems (29-31). Knobloch (42) and P e u s c h e l e t a l . (43) have o b t a i n e d somewhat b e t t e r agreement w i t h the use of Krupkowski's f o r m a l i s m f o r the a c t i v i t y c o e f f i c i e n t , w h i l e P a n i s h (32) has suggested t h a t agreement between the two s e t s o f paramet e r s may be o b t a i n e d by a d j u s t i n g the pure component vapor p r e s s u r e s o f phosphorus and a r s e n i c . The d i s c r e p a n c i e s between the calculated and experimental thermodynamic p r o p e r t i e s have been d i s c u s s e d by S t r i n g f e l l o w ( 4 1 ) , who c o n c l u d e d t h a t w h i l e the simple s o l u t i o n model i s a u s e f u l t o o l f o r the c a l c u l a t i o n o f phase diagrams, i t i s unable t o r e p r e s e n t the o t h e r thermodynamic p r o perties. A number o f o t h e r models have been used i n c o n j u n c t i o n w i t h E q u a t i o n 7 t o c a l c u l a t e the b i n a r y phase diagrams. Among these are the q u a s i c h e m i c a l e q u a t i o n (35^,44), a t r u n c a t e d Margules equat i o n (45,46), Darken's f o r m a l i s m (47,48), and v a r i o u s forms o f the c h e m i c a l t h e o r y , i n which a s s o c i a t e d l i q u i d s p e c i e s are p o s t u l a t e d and some assumptions are made about the p h y s i c a l i n t e r a c t i o n s between the s p e c i e s (49-51). S e v e r a l o f t h e s e s t u d i e s have cons i d e r e d the l i q u i d phase p r o p e r t i e s as w e l l as the l i q u i d u s i n the parameter e s t i m a t i o n (45,46,51). S o l i d S o l u t i o n Behavior. Attempts t o c a l c u l a t e t e r n a r y phase diagrams w i t h the use o f the a v a i l a b l e b i n a r y s i m p l e s o l u t i o n parameters have met w i t h f a i r s u c c e s s . I t i s necessary i n performing such c a l c u l a t i o n s t o a s s i g n a v a l u e t o the s o l i d s o l u t i o n activity coefficient i n E q u a t i o n 6. T h i s has usually been



290

a c c o m p l i s h e d by a f i t o f the pseudobinary phase diagram, i n which the l i q u i d phase mole f r a c t i o n of the Group I I I elements and Group V elements a r e r e q u i r e d t o each sum t o 0.5, o r by s e t t i n g the s o l i d s o l u t i o n a c t i v i t y c o e f f i c i e n t e q u a l t o one. The latter approach i s o n l y u s e f u l i n systems such as Ga A l ^ _ As i n which the l a t t i c e mismatch i s n e a r l y z e r o . The general resu¥ts o f such c a l c u l a t i o n s show t h a t the c a l c u l a t e d t e r n a r y l i q u i d u s i s o t h e r m s a r e i n f a i r agreement w i t h e x p e r i m e n t a l d a t a , and are i n s e n s i t i v e t o the c h o i c e o f a p a r t i c u l a r s e t o f parameters. The c a l c u l a t e d t e r nary s o l i d u s i s o t h e r m s , however, appear t o be more s e n s i t i v e t o the v a l u e s o f the i n t e r a c t i o n parameters, and the agreement i s f a i r t o poor f o r t e r n a r y I I I - V systems. Several investigators (33^3!?,38.) have a d j u s t e d the v a l u e s o f the b i n a r y parameters t o o b t a i n a b e t t e r r e p r e s e n t a t i o n o f the t e r n a r y system. As a s p e c i f i c example, G r a t t o n and Woolley (39) have measured s o l i d u s i s o t h e r m s i n the Ga-In-Sb system by a n n e a l i n g samples i n the two-phase f i e l d f o l l o w e d by r a p i d quenching. The s o l i d conc e n t r a t i o n s were then determined by x-ray a n a l y s i s . While the parameter s e t chosen by t h e s e a u t h o r s as w e l l as s e v e r a l o t h e r s (23,y7,38^ showed r e a s o n a b l e agreement w i t h the a v a i l a b l e l i q u i d u s i s o t h e r m s , the agreement w i t h the s o l i d u s i s o t h e r m s was poor i n each c a s e . In a d d i t i o n , each o f the parameter s e t s p r e d i c t s a p o s i t i v e e n t h a l p y o f m i x i n g over the e n t i r e c o m p o s i t i o n r e g i o n , whereas the e x p e r i m e n t a l measurements (52!>f^3) i n d i c a t e t h a t the e n t h a l p y o f m i x i n g i s n e g a t i v e f o r most c o m p o s i t i o n s . A s e l a g e and Anderson (26) have used a b i n a r y d a t a base that i n c l u d e d e x p e r i m e n t a l v a l u e s o f e n t h a l p y o f m i x i n g and component a c t i v i t i e s i n a d d i t i o n t o l i q u i d u s temperature t o e s t i m a t e b i n a r y s i m p l e s o l u t i o n parameters. The p r e d i c t e d t e r n a r y m i x t u r e p r o p e r t i e s and l i q u i d u s temperature were i n e x c e l l e n t agreement w i t h e x p e r i m e n t a l measurements. F i g u r e 6 compares the p r e d i c t e d 873K t i e l i n e s (dashed l i n e s ) t o the e x p e r i m e n t a l measurements (solid l i n e s ) o f G r a t t o n and Woolley (39^). The agreement shown between the c a l c u l a t e d and e x p e r i m e n t a l s o l i d u s i s r e a s o n a b l y good and r e p r e s e n t s a s i g n i f i c a n t improvement over p r e v i o u s a t t e m p t s t o c a l c u l a t e the Ga-In-Sb phase diagram u s i n g o n l y b i n a r y parameters. One o f the main d i f f i c u l t i e s w i t h u s i n g the pseudobinary phase diagram as a d a t a base f o r e s t i m a t i n g the s o l i d s o l u t i o n p r o p e r t i e s i s t h a t i t r e p r e s e n t s o n l y the h i g h temperature b e h a v i o r . For most a p p l i c a t i o n s , the lower temperature p o r t i o n o f the phase diagram i s i m p o r t a n t ( e . g . , s o l i d u s i s o t h e r m s i n the 700-900K temperature range f o r a n a l y s i s o f l i q u i d phase e p i t a x y , the p r e d i c t i o n o f m i s c i b i l i t y gaps i n the s o l i d solution). The temperature dependence o f the s o l i d s o l u t i o n Gibbs excess energy i s found t o be s e n s i t i v e t o the s o l u t i o n model and method o f d a t a r e d u c t i o n used. As an example, Chang e t a l . (54) s t u d i e d the Ga In-^_ Sb pseudobinary system. The Ga In-^_ Sb l i q u i d m i x t u r e was t r e a t e d e i t h e r as a t e r n a r y m i x t u r e o f G a , ¥n and Sb, the thermodynamic p r o p e r t i e s b e i n g e s t i m a t e d 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 o f GaSb and InSb, the thermodynamic p r o p e r t i e s c a l c u l a t e d from the s i m p l e s o l u t i o n model w i t h the parameters e s t i m a t e d from a f i t o f the pseudobinary phase diagram. For b o t h d e s c r i p t i o n s o f 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 e q u a t i o n


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1.4

600 550 500 450

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1.8

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F i g u r e 5. V a l u e s o f versus reciprocaT~temperature as c a l c u l a t e d b y t h e N R T L e q u a t i o n w i t h E q u a t i o n 9. R e d u c t i o n o f d a t a set: , liquidus temperature; , l i q u i d u s temperat u r e and Ga a c t i v i t y ; , l i q u i d u s temperature and enthalpy o f mixing; , l i q u i d u s t e m p e r a t u r e , Ga a c t i v i t y and enthalpy of mixing; , r e c o m m e n d e d v a l u e s o f 0Qasb*

Sb

F i g u r e 6. G a - I n - S b 8 7 3 K t i e l i n e s . ; experimental results (39).

; simple

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292


was used t o model the s o l i d s o l u t i o n b e h a v i o r and parameters were e s t i m a t e d from a f i t o f the pseudobinary phase diagram. Both treatments of the l i q u i d phase gave s t a n d a r d d e v i a t i o n s i n the l i q u i d u s and s o l i d u s temperatures w i t h i n the e x p e r i m e n t a l uncertainty. The variation o f the Gibbs excess energy o f t h e s o l i d s o l u t i o n w i t h t e m p e r a t u r e , however, was i n o p p o s i t e d i r e c t i o n s f o r the two d i f f e r e n t t r e a t m e n t s o f the l i q u i d phase. I t f o l l o w s t h a t the low temperature segment o f the t e r n a r y phase diagram is p r e d i c t e d d i f f e r e n t l y f o r each assumption. An Example: r c . i n the Al-Ga-Sb System. r term in calculating t e r n a r y phase

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A 1

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D i v i d i n g E q u a t i o n 4 by E q u a t i o n 3 g i v e s the s i m p l e r e s u l t : K

I C

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F

I C

exp(-0 )

-

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I C

S i n c e 6 ^ i s not a f u n c t i o n o f c o m p o s i t i o n , the s l o p e o f a p l o t o f K j ^ v e r s u s x^ a l o n g an i s o t h e r m i s p r o p o r t i o n a l t o T^. This p l o t can be compared t o the s t r a i g h t l i n e t h a t i s produced when the value of Tjp i s assumed e q u a l t o u n i t y , which o c c u r s when e i t h e r the l i q u i d ana s o l i d m i x t u r e s are both ideal s o l u t i o n s or the d e v i a t i o n i n the l i q u i d phase ( X J / Q ) i s the same as t h a t i n the s o l i d (/JQ)Shown i n F i g u r e 7 i s the AlSb d i s t r i b u t i o n coefficient plotted as a f u n c t i o n o f the Sb l i q u i d phase mole f r a c t i o n i n the Al-Ga-Sb system a t 823K. The e x p e r i m e n t a l v a l u e s o f ^ A ] ^ were t a k e n from the s o l i d u s and l i q u i d u s measurements o f Dedegkaev et a l . (55) and Chang and Pearson (56). The dashed line r e p r e s e n t s the d i s t r i b u t i o n c o e f f i c i e n t c a l c u l a t e d from E q u a t i o n 13 w i t h the v a l u e o f I ^ c u = 1 and the v a l u e s Q/\]qh s e l e c t e d by Chang e t a l . (_4). Tne s o l i d l i n e i s the r e s u l t o f e v a l u a t i n g E q u a t i o n 13 w i t h the v a l u e o f ^V^^ a g a i n u n i t y but w i t h the v a l u e s o f O ^ g ^ d e t e r ' p n e d from E q u a t i o n 7 and the f o l l o w i n g p r o p e r t y v a l u e s : a ( T ) a ^ ( T ) = 0.25, i . e . i d e a l s o l u t i o n , AH ( A l S b ) = 82.0 k J / m o l e j T ( A l S b ) = 1338K and AC ( A l S b ) = 10.83 J/mole K. I t i s observed from F i g u r e 7 t h a t , f i r s t , ^ t h e v a l u e o f T ^ i c ^ i s not u n i t y and, second, t h a t t h e r e e x i s t s a p a r t i a l c a n c e l l a t i o n o f the c o m p o s i t i o n dependence i n the l i q u i d phase a c t i v i t y coefficient product by t h a t found i n the s o l i d s o l u t i o n . The second o b s e r v a t i o n s u g g e s t s t h a t the l i q u i d and solid s o l u t i o n model s e l e c t i o n process s h o u l d be i n s e n s i t i v e w i t h r e s p e c t t o l i q u i d u s and s o l i d u s d a t a a l o n e . Indeed, the assumption o f i d e a l solution behavior i n both phases c l o s e l y p r e d i c t s the c o r r e c t d i s t r i b u t i o n c o e f f i c i e n t , y e t e x p e r i m e n t a l measurements o f the s o l u t i o n thermochemical p r o p e r t i e s c l e a r l y i n d i c a t e moderate n e g a t i v e d e v i a t i o n s from i d e a l b e h a v i o r . A 1

An " e x p e r i m e n t a l " v a l u e f o r TJQ can be a s s i g n e d by solving E q u a t i o n 13 f o r T j ^ and r e q u i r e s v a l u e s f o r 9 ^ , based on the e x p e r i m e n t a l p r o p e r t i e s d i s c u s s e d p r e v i o u s l y , and liquidus and solidus data. Shown i n F i g u r e 8 i s a p l o t o f 1 - ^ - ^ versus x ^ along s e v e r a l isotherms. I t i s seen t h a t r c i s nearly A 1

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Ternary Group III- V Semiconductor

ASELAGE ET AL.

Materials

293

F i g u r e 7. T h e A l S b d i s t r i b u t i o n c o e f f i c i e n t v e r s u s t h e S b l i q u i d phase mole f r a c t i o n a t 323K. , A ] S b = 1; , AlSb = 1 and a ^ ( T ) a | M T ) = 0.25; 0, m e a s u r e d v a l u e s ( 5 6 ) ; • , m e a s u r e d v a l u e d (55). r

—

0.8

_

0.6

_

CO

0.0

0.02

0.04

0.06

0.08

F i g u r e 3. V a l u e s o f l - r ^ e ^ c a l c u l a t e d w i t h E q u a t i o n 13 v e r s u s the l i q u i d phase mole f r a c t i o n a tv a r i o u s temperatures. T h e c a l c u l a t i o n u s e d t h e r e c o m m e n d e d v a l u e o f 0A-|$b ' p h a s e d i a g r a m d e t e r m i n a t i o n s : t y , 7 7 3 K ( 5 5 ;; O , 8 2 5 K ( 5 5 ) ;

Solid-Liquid Equilibrium in Ternary Group III-V Semiconductor Materials

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