Thermodynamic modeling and phase equilibria calculations of the quaternary Al-Ga-In-Sb Sastem

S e c t i o n I: Bas ic and Applied R e s e a r c h

Thermodynamic Modeling and Phase Equilibria Calculations

of the Quaternary AI-Ga-In-Sb System

R.C. S h a r m a and M. Srivastava Department o f Materials and Metallurgical Engineering

Indian Inst i tute o f Technology Kanpur, UP 208016 , India

(Submitted November 23, 1992; in revised form December 28, 1993)

Quasi sub-subregular solution model with additional ternary parameters, used by Sharma et al. to model the thermodynamic properties of the liquid phases in the ternary AI-Ga-Sb, Al-In-Sb, and Ga- In-Sb systems, has been extended to predict the thermodynamic properties of the liquid phase in the quaternary AI-Ga-In-Sb system. The (AIGaIn)Sb compound phase in the quaternary A1-Ga-In-Sb system is considered a quasi-regular solution of AISb, GaSb, and InSb compounds. Phase equilibria in the quaternary AI-Ga-In-Sb system are then calculated and compared with the limited experimental data available in the literature. The ternary AI-Ga-In phase diagram, required for the quaternary calculations, has also been modeled and calculated.

I n t r o d u c t i o n T h e r m o d y n a m i c M o d e l s

Sharma and Mukerjeel and Sharma and Srivastava 2.3 recently analyzed and calculated the Ga-In-Sb, A1-Ga-Sb, and Al-In-Sb phase diagrams, respectively. In the present work, phase equilibria calculations are extended to predict the quaternary A1-Ga-In-Sb phase diagram. For this, thermodynamic descrip- tion of the A1-Ga-In systemis also required. Very limited phase equilibria and thermodynamic data are available in the literature on the A1-Ga-In and A1-Ga-in-Sb systems. Girard et alA, 5 measured thermodynamic properties of the liquid phase in the A1-Ga-In system. No phase diagram determinations of the Al- Ga-In system could be found in the literature. In the quaternary AI-Ga-In-Sb system, Zbitnew and Woolley 6 showed that there is complete miscibility between A1Sb, GaSb, and InSb compounds, and an isomorphous pseudotemary phase diagram is formed between these compounds. They also estimated some tie lines for liquid plus solid two-phase equilibrium in the A1Sb-GaSb-InSb pseudotemary section; however, only compositions of these tie lines are given, as the temperature could not be estimated.

In the present work, A1-Ga-in phase equilibria are calculated by using a quasi sub-subregular solution modep-3 for the liquid phase. This model is then extended to predict the thermodynamic properties of the liquid phase in the quaternary A1-Ga- In-Sb system. The (A1GaIn)Sb compound is considered a quasi-regular solution of A1Sb, GaSb, and InSb compounds. Phase equilibria in the quaternary AI-Ga-In-Sb system are then calculated by using the optimized thermodynamic parameters of the constituent temary systems.

Ternary S o l u t i o n P h a s e s

Molar Gibbs of a ternary solution phase may, in general, be written as:

3 3

G= Z XiG~ + R Y E xilnmi + AmixGeX z = l t = l

(Eq 1)

where X, is the mole fraction of component i in the solution, G 0 is the molar Gibbs energy of component i in the reference state (pure component i in the same phase), R is the gas con- stant, Tis the temperature in kelvin, and A~xGex is the excess molar Gibbs energy of mixing. AmixGex for the ternary solution is expressed by a quasi sub-subregular solution modeP -3 as:

Table 1 Lattice and Compound Stabilities

GOAIL_ GOAI s = 10 7 9 2 . 0 - 1 1 . 5 6 T J / m o l

G ~ a L - G ~ a s = 5 5 8 9 . 8 2 - 1 8 . 4 5 2 9 T J / m o l

G~lnL - ~iin s = 3 2 6 3 . 5 0 - 7 . 5 9 3 4 T J / m o l

,L ,S _ ~SSb - ~ S b - 19 8 7 4 . 0 - 2 1 . 9 8 6 8 T J / m o l

G'~ 'L G~. ' f c c - 5 0 2 1 . 1 8 - 8 . 3 6 9 9 T J / m o l C - a - Ga - -

G~Ssb - G~ s - ~ss~ = - 4 7 6 4 7 . 0 5 6 - 3 6 . 6 5 1 8 4 T + 6 . 4 9 7 7 T I n T J / m o l

G?'SGasb - GO.SGa -- G~s'b s = - 41 2 4 0 . 5 - 1 . 3 9 4 1 T + 1 . 8 4 2 6 T l n T J / m o l

,s G~. ' s - G-~ 's = - 3 0 2 5 2 . 3 - 0 . 2 4 6 T + 2 . 4 2 8 4 T l n T J / m o l G~InSb-- In Sb

178 Journal of Phase Equilibria Vol. 15 No. 2 1994

B a s i c a n d A p p l i e d R e s e a r c h : S e c t i o n I

Table 2 Optimized Parameters for Liquid Phase (I:AI, 2:Ga, 3:In, 4:Sb)

w(1,2) = 2 4 9 . 0 3 / T - 0 . 1 0 1 2 4 w(2,1) = 4 0 4 . 8 3 / T - 0 .25633 v(1,2) = 0

w(1,4) = - 1 9 3 2 . 9 3 / T + 0 .3196 w(4,1) = 6 0 7 . 9 4 / T - 0 .03867 v(1,4) = 704 .3 I/T

w(2,4) = - 1 6 5 7 . 6 / T + 9 .6150 1.2631 l nT w(4,2) = - 2 2 2 1 . 6 / T + 15.3011 96 2 .0160 lnT, w(2,4) = 2 1 3 . 9 / T + 0.0435

w(3,4) = - 1 9 4 6 . 6 / T + 9.1611 - 1 .19601nT w(4,3) ~ - 3 4 6 9 . 1 / T + 15.431 6 1 .96201nT v(3,4) = 3 0 0 . 3 / T + 0 .3230

u123(1) = - 2 3 2 3 . 6 / T + 0 .71627

u 123(2) = u 123 (1) Ul23(3) = u123(1)

u134(1) = - 5 2 6 1 . 6 7 / T

U134(3 ) = U134(1) U134(4) = U134(1)

w(1,3) = 2 7 5 5 . 1 / T - 0 .132 w(3,1) = 3 8 2 4 . 2 / T - 0 .606 v(1,3) = 4 3 9 . 1 / T - 0 .086

w(2,3) = 533 .55 /T + 0 .1315 w(3,2) = w(2,3) v(2,3) = 0

u124(1 ) = 6 3 3 4 . 6 1 / T - 7 .0845 ul24(2 ) = - 3 0 0 8 . 2 4 / T + 6 .7068 u124(4 ) = - 7 7 0 9 . 4 4 / T + 3 .6247

u234(2 ) = - 6 5 1 . 3 4 / T - 2.45 88

u234(3) = u234(2) u234(4) = u234(2)

N o t e : Subscriptij, kinuokrefertouparametersfortheternarysystemconsist- ing o f c o m p o n e n t s i, j , and k.

Table 3 Optimized Parameters for AI-Ga fcc Solid Solution (I:Ai, 2:Ga)

wfeC(1,2) = - 692 .93/T + 1.6092 wfCC(2,1) = wfCr

Ve~e2c (1,2) = 0

A~x Gex

RT 2

Z l = l

Z wj, -

J = / + l ~

+ X1X2X 3 (uIX 1 + uzX 2 + uaX3) (Eq 2)

with the different solution parameters, w~ vif, and u i, in general, expressed as: 1,3.7

w 0 - ~ + B v + C,jinT (Eq 3a)

D v i j : T + Eij (Eq 3b)

M i u : - - + N~ (Eq 3c)

T where A,j, B O, C,j, Dq, Eij, M i, and Ni are constants. Normally the temperature dependence of solution model parameters is expressed as A/T + B, which corresponds to temperature inde-

pendent enthalpies and excess entropies of mixing. However, here A/T + B + C lnTtype of temperature dependence is used for some of the parameters to account for temperature depend- ent enthalpies of mixing found in Ga-Sb and In-Sb systems.7 Now, the partial molar Gibbs energy, G~p x, of a component, p, in the solution can be obtained by standard methods and is given by:l-3

In -wiP + w pi

l = 1(1 ~p)

- Z Z wv wj, +wo-wJ,)(Xj-X,)-12v,jXiXj X~j

t 1 = J = t + l

2 3

+ Z Z XjXk[2UpXp+ ujXj + ukXk] j = I q ~ p ) k=j+l(k~:p)

- 3X1X2X3[ulX 1 + u2X 2 + u3X3] (Eq 4) where yp is the activity coefficient of component p in the solution, and vij = Vir The partial molar Gibbs energy (chemical po- tential), Gp, of component p is then obtained as:

Gp = G~p + RTlnXp + RT lnTp (Eq 5)

Equations 1 to 5 describe the thermodynamic properties of the ternary solutions. These equations reduce to corresponding equations for binary solutions when concentration of one of the components goes to zero.

Q u a t e r n a r y S o l u t i o n P h a s e s

The thermodynamic model given above for ternary solutions is extended to quaternary solutions without any additional quaternary interaction parameters. Molar Gibbs energy of the solution is then given as:

4 4

G= Z X iG ~ + RT Z X i lnX + AmaxGeX (Eq6) i = 1 s = l

where all terms are as defined earlier, and excess molar Gibbs energy of mixing is given as:

3 4 ~ - x G e x + W . . - - W..

RT -~-~ z[wiJ2 ~i + ( ~ ~ ( X ' - X " 2 "'--J " i=1 j = i + l

2 3 4

- Z Z Z XiXjXk [u(i,l )X i + u(j,l)Xj

i= l j=t+l k=j+l

+ u(k,l)X k] (Eq 7)

with

l = i + j + k - 5 (Eq8) where I identifies the ternary interaction parameters u (i,/), u(j,/), and u(k,1) corresponding to the ternary solution

Journal of Phase Equilibria Vol. 15 No. 2 1994 179

S e c t i o n I: Bas ic and Appl ied R e s e a r c h

Table 4 Optimized Parameters for AlSb-GaSb-InSb Solid Solution

(I:AISb, 2:GaSb, 3:InSb) o~(1,2) = ct(2,1) = 5 21 .65 /T - 0.5142 ct(1,3) = ct(3,1) = 433.67/T ct(2,3) = ct(3,2) = 510.85/T + 0.6307

of components i , j , and k. The excess partial molar Gibbs energy, C~pX, of a component, p, is now given as:

ex 4 I 2 ~ t = lnTp = E w p Wpi + (WtP _ Wp, ) (Xp - ~ ) - 8"dijXpX i

R T t = 1 (tO:p)

-,:__~ ,=i+1 ~ - - t- (w v - wji)(X j - X ) - 12vqX XtX J

3 4

+ X X t2u o, x + u ,0x / = l (lap) k=j+l (kg:p)

2 3 4

+ u(k,l)Xk]Xflk- ~, ~, ~, 3xix~Xk[u(i,1)xi l = l j = i + l k = j + l

+ u(j,l)Xj + u(k ,1 )Xj (Eq 9)

with I = (i + j + k - 5) = (p + j + k - 5) and vij = vii. T h e partial molar Gibbs energy, Gp, of component p is then given by:

Gp : G~p + R T lnXp + R T lnTp (Eq 10)

AlylGayzInynSb P h a s e

A1Sb, GaSb, and InSb compound phases are essentially stoichiometric and are here considered as line compounds. Their Gibbs energy of formation as a function of temperature may be expressed as: 1-3

GOMOsb - G~ ~ - G~s" ~ = A + B T + C T lnT (Eq 11)

where M stands for AI, Ga, or In; 0 stands for the compound phase; and A, B, and C are constants. A1Sb, GaSb, and InSb are completely miscible, and their solid solution may be represented as (AI r Ga r Inr )Sb, where Y1, Y2, and ]13 a le mole fractions of ~lSb 2, ~aSb, and InSb, respectively, and Y1 + Y2 + Y3 = 1. In the present investigation, 0 phase, i.e. (Alv Gay Inv~ )Sb solid solution, is considered a quasi-regular

1. 2 ~ sotutaon of A1Sb, GaSb, and InSb compounds with no pseudotemary interaction parameters. Accordingly, molar Gibbs energy of the 0 phase is expressed as:

3 3

G~ Yi6*+RT~ Y, InY+A =G ~ (Eq 12)

t = l ~=l

where i = 1 to 3 stands for components AISb, GaSb, and InSb, respectively, and the excess molar Gibbs energy of mixing, A=xGeX, is given by:

AmixGe x 2 3

R T - ~ E tx(id')Y'YJ (Eq 13) i=1 j = t + l

with the solution parameters tx(ij) expressed as:

A IJ "4-BIj (Eq 14) a( i d) : -f

where A'tj and BIj are constants. The excess partial molar Gibbs energy, C~pX, of any component is then given by:

Gpx 3 2 3

R--T = E ~(P'i)Yi- E E o~(id)Yyj (Eq 15) i=l(i~tp) l=1 j = i + l

where ~(id) = tx(j,i). T h e partial molar Gibbs energy of any component, p, in the 0 phase is now given by:

G~p = G~/~ + R r lnYp + G p x (Eq 16)

Phase Equilibria

Ternary ALGa-In S y s t e m

Pure A1, Ga, and In melt at 933.602,302.9241, and 429.784 K, respectively. Binary A1-Ga and Al-In systems were recently assessed and calculated by Sharma and Srivastava 2,3 using the same thermodynamic models as described in the previous section applied to binary systems. Ga and In melt at relatively low temperatures, and phase equilibria here are calculated at temperatures higher than the melting points of Ga and In. There- fore, calculation of the binary Ga-In system is not considered here. AI and Ga form a simple eutectic phase diagram with the eutectic close to the melting point of Ga at -97.6 at.% Ga. A1 dissolves -8.6 at.% Ga in solid solution, whereas the solubility of AI in solid (Ga) is negligible. In the Al-In system, there is a liquid miscibility gap with a critical point at -1182 K, a monotectic reaction at -912.6 K, and a eutectic close to the melting point of In. Solid (AI) dissolves a maximum of -0.4 at.% In at the monotectic point, and the solubility of Al in (In) is negligible. For phase equilibria calculations in this study, solubility of In in (A1) is also considered as negligible. In the A1-Ga-In system, therefore, the liquid miscibility gap starting from the Al-In binary and the equilibrium between Al-rich Al- Ga solid solution and liquid phase are of interest at temperatures higher than the melting points of Ga and In.

Equilibrium between L I and L 2 within the liquid miscibility gap in the ternary system is given by:

G~/1 = G~2 (i = AI, Ga, an) (Eq 17a,b,c)

For L 1 + L 2 equilibrium, there are four compositional unknowns, two for each phase. By fixing one of the compositional variables, Eq 17a to c are simultaneously solved to give one tie line in the L 1 + L 2 region. Calculation of the number of tie lines defines the complete miscibility gap. The critical point of the miscibility gap, at a given temperature, is given by: 7


Basic and Applied Research: S e c t i o n I

-~22 + 3nLo~z~X3) 5nL~x~-2~+ n L ~ 3 ~ = 0 (Eq 18b)

where G is molar Gibbs energy, X 2 and X 3 are molar fractions of components 2 and 3 in the liquid, and n is given by:

aZG/'OX~z 32G/'OXzOX 3 n . . . . (Eq 18c)

c)2G/CO2aX 3 ~2G/0X~3

Equations 18a and b are simultaneously solved for composition of the critical point at a given temperature. Detailed equations for partial derivatives in Eq 18a and b are given in reference 3.

Equilibrium between AI-Ga solid solution and the liquid phase, L 1 or L z, in the A1-Ga-In system is given by:

G~,] = %(L2) (Eq 19a)

and

G~a = GLG~L2 ) (Eq 19b)

where (x stands for Al-rich fcc solid solution of Ga in A1. It is assumed that the solubility of In in fcc A1 is negligible. Now, for these two phases in equilibrium, there are only three unknowns, one for the fcc solid solution and two for the liquid phase. By fixing one, Eq 19a and b are solved for the other two to give one tie line, and calculation of the number of such tie lines defines the whole ot + L 1, or (x + L 2, equilibrium region. Interaction between L I + Le, (x + L1, and (x + L2 regions may give rise to a three-phase equilibrium triangle and can be solved by combining Eq 17 and 19.

Quaternary AI-Ga-In-Sb S y s t e m

At temperatures higher than the melting points of Ga and In, the quaternary A1-Ga-In-Sb system consists of the following phases: (1) liquid phase with a liquid miscibility gap, L 1+L2, in some regions; (2) (A1 r Gay Inr )Sb compound i.e. 0 phase;

�9 2 3 . �9 .

(3) Al-rich A1-Ga fcc sohd solution, say ct; and (4) solid Sb with no solubility for other components. Equilibrium between the liquid phase and 0 is given by the following thermodynamic equations:

+ = lSb (Eq 20a)

GLGa + OLb = GOGaSb (Eq 20b)

G~x, + C~Sb : ~II~Sb (Eq 20c)

In the L + O two-phase equilibrium, there are five compositional unknowns, two for the O phase and three for the liquid phase. A single tie line in the (L + O) region may be calculated by fixing two of the compositions in one of the phases and solving the others by simultaneous solution ofEq 20a, b, and c. Since a section of the quaternary phase diagram through 50 at.% Sb is a true pseudotemary section, pseudoternary isotherms of the A1Sb-GaSb-InSb pseudotemary diagram may be calculated by fixing X~s b = 0.5 and using Eq 20a, b, and c.

The equilibrium between L 1 and L z in the miscibility gap is given by:

G~, ~ = G~/2 (i = A1, Ga, In, Sb) (Eq 21a, b, c, d)

Here, for the two phases in equilibrium, there are six compositional unknowns, three for each phase. Again, two variables

6000 I I I I /

5000 1 ~ 9 / l

2 o 3 / I

• 6 x 1/3

1 <~ 7 ~ 1/9

000 0 ~ 0 0.2 0.4 0.6 0.8 1.0

X I n

Fig. 1 Comparison of calculated molar cnthalpy o f mixing in AI-Ga-In alloys wi th the experimental data of Girard et al. 4,5 measured at 990, l l l2 , and 1182K.


S e c t i o n I: B a s i c a n d A p p l i e d R e s e a r c h

14 i j J i

12 ~% XGe/XIn

I o I/3

\ 2 o I 10

~ % 3 �9 3

7 ~ 4 + 9

�9 3 ,

2

| + '

0 ! I 1 I ~ 1 0 0.2 0 4 0.6 0.8 1.0

XA[

Fig. 2 Comparison of calculated partial molar excess Gibbs energy of A1 in liquid A1-Ga-In alloys at 1000 K with the experimental data of Girard et al 4, 5.

15000

XGa / Xl n

12500 - ~ I o I / 3

2 ~, I

3 �9 3 10000 - 4 + S

,~176176 ,

2500 - 4

I I i I 0~ 0.2 0.4 0.6 0.8 1.0

XA L

Fig. 3 Comparison of calculated pardal molar enthatpy of A] in liquid A1-Ga-In alloys at 1000 K with the experimental data of Gi- rard et al 4,5.

In

T = 850 K

Fig. 4

0 O1 0.2 0.3 0.4 A[

Calculated A1-Ga-ln isothermal section at 850 K.

0.5 0.6 0.7 0.8 0_9 1.0

X6o GO

for one of the phases are fuxed, and Eq 21a to d are solved for the other four to give one tie line o f the L 1 + L2 region. Calcu- lations can be repeated to cover the whole miscibil i ty gap.

The equi l ibr ium be tween A I - G a fcc solid solution and liquid is given by:

GT~ 1 = GLA1 (Eq 22a)


Basic and Appl ied Research: S e c t i o n I

0 0.1 0.2 0.3 0.4 O.b A| x6a

In

0.(5

Calculated A1-Ga-In isothermal section at 750 K.

u. / 0.8 0.9 1.0

Ga

T = 750 K

Fig. 5

In

Fig. 6

AI XG a

Liquidus projections of the A1-Ga-In phase diagram (calculated).

,0 Ga

and

(Eq 22b)

where ~ stands for fcc solid solution, and solubilities of In and Sb in ~ are assumed to be negligible. In this case, there are four unknown composit ional variables, one for u and three for the


S e c t i o n I: B as i c a n d Appl i ed R e s e a r c h

Fig. 7

O0 0.1 0.2 0.3 0.4 GaSb

AISb ' I N

Calculated AlSb-GaSb-InSb isothermal section at 1123 K.

0.5

YlnSb 0.6 O.7 0.8 o.9

\ 1.0

InSb

Fig. 8

0.2

0.1

Oo/ 0.1

GoSb

0.6

"kv 0.5

0.4

0.3

O.-q

0.8

0.7~/~

0.2 0.3 0.4

Calculated AlSb-GaSb-lnSb isothermal section at 873 K.

AlSb 1.0^

0.5 YInSb

T=873 K

0.6 0.7 0.8 0.9 L/" 1 0 InSb

liquid phase. By fixing two of these compositions, Eq 22a and b are simultaneously solved for the other two to give one tie line, and the number of such tie lines are calculated to define the whole region.

For equilibrium between solid Sb and the liquid phase, there is only one thermodynamic equation:

__ % (Eq 23)


B a s i c a n d A p p l i e d R e s e a r c h : S e c t i o n I

$b

I , 1'~7q K

2 3 4 5 6

7 cl7"~ K AISb

i N

ey

O0 0.I 0.2 0.3 0.4 0.5

GoSb YlnSb

0.6 0.7 0.8 0.9

1 .I0

1.0 InSb

Liquidus and solidus projections of the A1Sb-GaSb-InSb pseudotemary phase diagram (calculated).

b = 6/z.

Fig. lo

Fig. 9

I

0 o 0.1 0.2 0.3 0.4 0.5 0.6

XGo+Xln

0.7 0.8 0.9 1.0

0 + L equilibrium in the AI-Ga-In-Sb system at 1123 K for l'~aSb/~Sb ratio of 6/4 (calculated).

and (Sb) liquidus is obtained by fixing the composition of two of the components in the liquid phase and solving the third by using Eq 23.

Finally, equilibrium between tx and 0 may be solved as fol- lows. Equations similar to 20a to c may be written for equilibrium between tx and 0 by substituting tx for L. However, since


Sec t ion I: Basic and Applied Research

Fig. 11

Sb

= ~/6

00 0.1 0.2 0.3 0.~ 0.5 0.6 0.7 0.8 0.9 1.0 A[ xGa + Xtn

0 + L equilibrium in the Al-Ga-In-Sb system at 1123 K for ]'~aSb~nSb ratio of 4/6 (calculated).

cc has no solubility for In and Sb, Gfn and Gg 5 are not known. Therefore, equations corresponding to 20a and b are combined to give:

c~ 0 ~ - 0 GOGaSb (Eq 24) G A I - - GGa - G A I S b - -

For c~ + 0 equilibrium, there are three unknown compositions, one for the (x phase and two for the 0 phase. Two of these are fixed, and Eq 24 is solved to give one tie line. Then, the calculations are repeated for other tie lines in the c~ + 0 region.

In the quaternary system, the above-mentioned, various two- phase regions may interact to give three-phase and four-phase equilibrium regions at a given temperature. They may be solved by combining the appropriate equations.

E v a l u a t i o n o f M o d e l P a r a m e t e r s

Table 1 lists the lattice stability data for pure components and Gibbs energies of formation for AISb, GaSb, and InSb compounds required for phase equilibria calculations in A1-Ga-In and AI-Ga-In-Sb systems from previous evaluations by Sharma et al.1.2,3,7 A1-Ga and AMn binary systems were evalu- ated recently by Sharma and Srivastava 2,3 using the same models. Their optimized parameters for the liquid and solid phases in these systems and that of Sharma and Mukerjee 1 for liquid Ga-In are used in this work and are given in Tables 2 and 3. Now the only parameters that need to be known to solve phase equilibrium in the A1-Ga-In system are the ternary parameters-u1, u2, and u3---in Eq 2 for the liquid phase. These are obtained by optimization with respect to the thermodynamic data of Girard et al. 4,5 Optimized values of these parameters are also given in Table 2.

Thermodynamic models for different phases in the A1-Ga-In- Sb quaternary system, as described in the section "Thermody- namic Models," do not use any additional parameters other than those obtainable from constituent binary and ternary systems. For calculation of phase equilibria in the A1-Ga-In-Sb system, parameters corresponding to the AI-Ga-Sb, Al-In-Sb, and Ga-In-Sb systems are taken from Sharma and Srivastava 2,3 and Sharma and Mukerjee,1 respectively. These, along with the parameters for AI-Ga-In from this study, are given in Tables 2 to4 .

R e s u l t s a n d D i s c u s s i o n

AI-Ga-In S y s t e m

Figure 1 compares the calculated molar enthalpy of mixing, and Fig. 2 and 3 compare, respectively, the calculated partial molar excess Gibbs energy and partial molar enthalpy of mixing of AI in A1-Ga-In alloys with the experimental data of Gi- rard et al.4,5 The agreement, in general, is quite satisfactory. In Fig. 1, the residual values atxm = 0 are the molar enthalpies of mixing of binary AI-Ga alloys at given atomic ratios. No phase equilibria determinations in the A1-Ga-In system could be found in the literature. The phase diagrams given here are based on our calculations. Figures 4 and 5 show the calculated isotherms of the AI-Ga-In phase diagram at 850 and 750 K, respectively. Below the critical temperature of the liquid miscibility gap in the binary Al-in system, 1181.9 K, a liquid miscibility gap appears in the ternary AI-Ga-In system. As the temperature falls below the melting point of A1, 933.602 K [Massalski2], A1-Ga solid solution will be in equilibrium with L 1 in the Al-rich comer. Further below, at temperatures lower


Basic and Applied Research: S e c t i o n I

60l/ I I I ~ t

�9 I I , 6 /6 /..0~ / 2 ~16

o I / ' i

20 2

1.0

(]0 02 01, 06 08 10

Fig. 12 X~a/X~, ratios along the liquidus in Fig. 10 and I1 as a function

-

. . . . i

than the monotectic temperature (912.6 K) in the Al-In system, (AI)+L 1 equilibrium interacts with the LI+L 2 equilibrium to give (A1)+LI+I_ ~ three-phase equilibrium as seen in the 850 K isotherm in Fig. 4 With further decrease in temperature, the miscibility gap in the ternary system sinks below the (A1) liquidus, the three-phase equilibrium triangle vanishes, and only two-phase equilibria between the Al-rich Al-Ga solid solution and liquid exist, as seen in the 750 K isotherm in Fig. 5. At the temperature at which the three-phase triangle just vanishes, the (A1) liquidus just touches the metastable liquid miscibility gap at its critical point. The calculated values of this point are xGa = 0.2897, Xln = 0.2803, and T = 786.46 K. At still lower temperatures, solid In and solid Ga phases will appear in the system Figure 6 gives the liquidus projections of the Al- Ga-In phase diagram.

AI-Ga-In-Sb S y s t e m

There are hardly any phase equilibria determinations in the A1- Ga-In-Sb system. Hence, the phase diagrams given here are predictions based on our calculations. In the quaternary A1- Ga-In-Sb system, a pseudotemary phase diagram, AISb- GaSb-inSb, with complete miscibility in liquid as well as solid exists. 6 Figures 7 and 8 give the calculated isotherms of the AISb-GaSb-InSb pseudotemary phase diagram at 1123 and 873 K, respectively. Figure 9 shows the liquidus and solidus projections of the AlSb-GaSb-inSb phase diagram at various temperatures. Also shown in Fig. 9 are a few tie lines for the liquid plus solid equilibrium estimated by Zbilnew and Wool- ley. 6 Temperatures of these tie lines are not known. They generally seem to agree with the calculated diagram.

In the quaternary AI-Ga-in-Sb system, various two-phase equilibria, i.e., the liquid miscibility gap and phase equilibria between 0 and liquid, a (Al-rich A1-Ga solid solution) and liquid, and (Sb) and liquid, can be calculated by using equations given in the section "Phase Equilibria." To graphically repre- sent isothermal sections of the quaternary diagram, three-dimensional plots would be required, which are generally difficult to read quantitatively. A quaternary isothermal section may, however, be plotted as a series of two-dimensional plots, where phase equilibria information is obtained by reading two

or more graphs in conjunction with each other. As an example, liquidus of the 0 phase in Al-Ga-In-Sb system at 1123 K at FI~l~b/I~s ~ ratios of 6/4 and 4/6 is plotted in Fig. 10 to 12. In

10 ~/fi~l 11, 0+L equilibrium is plotted on a A1-Sb-(Ga+In) triangle, along with the tielines for ~aSb/l~n ratios of 614 and InSb 4/6, respectively, and in Fig. 12, the corresp9_ tiding ~e~t- , ratio of the liquidus is plotted as a function of ~b. At a fixed temperature, 0+L equilibrium has two degrees of freedom. When atomic ratio Ga to In in the 0 phase is fixed, say at 6/4, then 0 end of the 0+L tie line at a given value of X~a (or :Kin) can be read from Fig. 10. In 0 phase, X~b is always equal to 0.5 and when X~/S~, and X~a (or X~n) are known, its composition is completely fixed. The composition of the liquid end of the 0 + L tie line as ~ and (X~a+XI'n) can then also read from Fig. 10 andX L/XoInL atX~ is obtained from the appropriate curve in Fig. 12,~us completely defining the composition of the liquid end of the tie line also. Similar plots at various I~S~nsb ratios at 1123 K would then completely describe 0 + L equilibrium at 1123 K. Other two-phase equilibria at different temperatures can be similarly calculated and represented by judicious choice two-dimensional plots. Three-, four-, or (in- variant) five-phase equilibria in the quaternary AI-Ga-In-Sb system can also be calculated by combining appropriate equations given in the section "Phase Equilibria" and represented in the similar manner. The choice of coordinates of the two-dimensional plots used for representing different phase equilibria would depend on the stability range of the different phases involved.

C o n c l u s i o n

A1-Ga-in and A1-Ga-In-Sb phase diagrams were calculated using quasi sub-subregular solution models with additional ternary interaction terms for the liquid phase and a quasi-regular solution model for the compound phase. The calculated thermodynamic properties for the liquid phase in the A1-Ga-In system agree well with experimental data in the literature. The Al-Ga-In and AI-Ga-In-Sb phase diagrams were calculated over the range of temperatures and compositions and pre- sented in graphical form.

A c k n o w l e d g m e n t

Authors gratefully acknowledge the financial support pro- vided by the Department of Electronics, Government of India, New Delhi.

Cited References

1. 1LC. Sharma and L Mukerjee, J. Phase Equilibria, 13, 5-15, (1992). 2. ILC. Sharmaand M. Srivastava, Calpha~ 16(4), 387-408 (1992). 3. R.C. Sharmaand M. Srivastava, Calpha~ 16(4), 409-426 (1992). 4. C. Girard, R. Baret, and J.P. Bros, J. Chent Thermodyn-, 10, 289

(1978). 5. C. Girard, J.P. Bros, and M. Hoch, Ber. Bunsenges. Phys. Chent, 92,

745 (1988). 6. K. Zbitnew and J.C. WooUey, J. AppL Phys., 52, 6611 (1981). 7. R.C. Sharma, T. Leo Ngai, and Y.A. Change, J. Electron. Mater., 16,

307 (1987). 8. J.L. Meijering, Phih'psRes. Rep., 3, 333 (1950).


Documents

Thermodynamic modeling and phase equilibria calculations of the quaternary Al-Ga-In-Sb Sastem