Download - Temperature dependence of associated liquid species, heat capacity and enthalpy of mixing for CdTe solution growth

ELSEVIER Journal of Crystal Growth 171 (1997) 525-530

, . . . . . . . . C R Y S T A L O R O W T H

Temperature dependence of associated liquid species, heat capacity and enthalpy of mixing for CdTe solution growth

X. Ye *, B. Tabarrok, D. Walsh Department of Mechanical Engineering, and Centre for Advanced Materials and Related Technology University ~f Victoria, Victoria, BC,

Canada, V8W 3P6

Received 26 March 1996; accepted 3 July 1996

A b s t r a c t

The mole fractions of liquid species, solution heat capacity, and enthalpy of mixing in a Te-rich liquid for CdTe crystal growth are calculated using the full associated liquid model for the liquid phase. The calculation results have shown that CdTe is the major solute species in the liquid and temperature effect on its dissociation is small over the temperature range of interest. Temperature dependences of the liquid heat capacity and enthalpy of mixing for different atomic compositions have also been given, which provides information on determination of growth temperature and understanding of the liquid state. The calculations indicate that design of a preheating process of the liquid may be important for high quality crystal growth of CdTe material.

1. I n t r o d u c t i o n

CdTe crystal has been grown from a melt (Bridg- man method) and a Te-rich solution (traveling heater method). However, experimental evidence [1,2] has shown that the growth kinetics is influenced by incorporation of the associated liquid species (CdTe) affecting the crystal structure and orientation. The I I -VI CdTe material system involves strong chemical interaction between the components of the compound [3,4]. A preheating process to dissociate the liquid compound before crystal growth may be important because certain elemental species present in the liquid is essential for high quality crystal growth [1,2]. In the case of CdTe grown from a Te-rich

* Corresponding author. Fax: +1 250 721 6051; E-mail: [email protected].

solution, we need to know the temperature-mole fraction relations for the liquid species in the solution. There is some concern on the temperature sensitivity of the liquid CdTe dissociation. On the other hand, for simplified modeling of mass transport in the Te solution growth process [5], we need to determine the main solute in the solution. Then the following questions arise: Is the liquid CdTe the major solute in the Te solvent? What about the relative amount of the other solute (Cd)? Are these liquid species sensitive to temperature change? These considerations require us to know about the mole fractions of the liquid species and their temperature dependence. In the following we present a thermodynamic analysis to determine the temperature-mole fraction relations of various liquid species for CdTe grown from a Te solution, used in the traveling heater method. Since the heat capacity and enthalpy

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526 X. Ye et a l . / Journal of Crystal Growth 171 (1997) 525-530

of mixing are important liquid properties for crystal growth, we calculate them by solution thermodynamic analysis using the full associated solution model. This model has provided a quantitatively satisfactory fit to the phase diagram and thermodynamic data for the Cd-Te system [6]. Therefore, we believe that this model can predict reasonable esti- mates for the heat capacity in the liquid. The heat capacity is an important parameter in the thermal modeling of CdTe growth. The temperature dependence of the heat capacity and enthalpy of mixing will be discussed in Section 3. The calculation leads to a good understanding of the solution state.

2. Associated solution model for the liquid phase

The liquid solutions for crystal growth can be present in dissociated form (ideal solution) or associated form, depending on the material system. The nature of the Cd-Te liquidus curve indicates a strong chemical interaction between Cd and Te in the liquid state. Jordan has shown that the equilibrium constant for the dissociation reaction (CdTe ~ Cd + Te) is very small (~ 0) due to the small radius of the curvature of the liquidus curve near the CdTe melt- ing temperature (see Fig. 1). Therefore, the liquid phase is strongly associated, retaining the properties of CdTe compounds in the liquid [3]. Referring to Su [4], we use the full associated solution model and solution thermodynamic analysis to calculate the mole fractions of the liquid species.

1100

9 0 0 u c

P 2 7o0

E 2

5 0 0

3 0 0 0

443-~5"C

322"-2"C

20 40 60 80

Atom "/, Te

Fig. 1. The C d - T e phase diagram [8].

9 8 %

100

For the full associated solution model, in the case of a Te-rich solution for CdTe growth, the liquid phase consists of the species Cd, Te, and CdTe. If nca and nTo represent the number of gram-atoms of component element Cd and Te, respectively, while NCd, NTe and NCdTe represent the number of moles of liquid species Cd, Te and CdTe, respectively, then the conservation of atom types requires

nca =Ncd + Ncove, (1)

rite = Nve + NCdT~. (2)

Divided by the total mole number, N, (N =Ncd + Nve + NCdTe), Eqs. (1) and (2) become

= X c d - - xT Yc Te, (3)

YTe = XTe -- XcdYcdTe, ( 4 )

where Xco and XT¢ are the atomic fractions of components Cd and Te respectively (Xcd = nco/(nco + nT~), XTe = nTe/(ncd + riTe))" YCd, YT~ and YCdTe are the mole fractions of liquid species Te, Cd and CdTe respectively (Ycd =Ncd/N, YTe = NTe/N, rCdTc = NCdTe/N)"

Under known atomic fractions XTe for preparing the initial solution at a given growth temperature, the species mole fractions YCd, YTe and YCdTe may be calculated using Eqs. (3) and (4). The third equation required to solve for the three unknowns can be obtained from the solution thermodynamic analysis as follows.

The Gibbs energy change upon mixing nCd of Cd and riTe of Te to form a homogeneous solution is given by

AGm = ncd ~cd(element) +nTe ~Te(element ), (5)

where ~cd(element) and ~Te(element) are the relative chemical potentials of component elements Cd and Te, respectively, i.e. the difference between the chemical potentials of the element in the solution and that of the pure liquid element. Using Eqs. (l) and (2) to eliminate ncd and riTe, Eq. (5) becomes

AG m = NCd ~cd(element) + NTe ~T~(element)

+ NCdT~(~cd(element) + ~Te(element)). (6)

X. Ye et al./ Journal of Cr)'stal Growth 171 (1997) 525-530 527

Therefore, the relative chemical potentials of the species become

(lAG m ~cd(species) -- - - -- ~cd(element), (7)

~Ncd

0AGm ~,Te(species) -- 0NT----- ~ -- ~Te(element), (8)

0AG m ~CdTe(Species) -- - - ~cd(species)

()NcdTe

+ ~ w e ( s p e c i e s ) . ( 9 )

Neglecting the effect of liquid mixing, the relative chemical potentials of the species can be expressed in terms of activity coefficients [6],

~Cd = Rg T l n ( 'YCd YCd ) , (10)

~We = RgT ln(Twe YTe ) , ( l l )

~CdTe = RgT ln(TCdTeYCdTe) -- AGOdT e, ( 1 2 )

where AGc0dTe is the Gibbs energy of dissolution of species CdTe into liquid elements Cd and Te, Rg is the universal gas constant and T is the absolute temperature. The chemical activity coefficients of species Cd, Te, and CdTe are denoted by TCd, YTe and "/COTe, respectively. Combining Eqs. (9) to (12), upon rearrangement, we obtain

YcaYT.______~ ~ _ TCdT_____~e ex p

YCdVe "YCd ]/Te Rg T

Substituting Eqs. (3) and (4) into (13), the following quadratic equation for YCdTe is obtained: [ (0)]

- AGCdTe (XcdXTe)YCdTe-- X2d + X2e -I- ~a ex p ~ ~---~

X YCdTe q- (XTcXCd) = 0 . ( 1 4 )

Solving Eqs. (14), (3) and (4), we can obtain the mole fractions, YCdT¢, Yca and YTe for a solution of interesting composition. An iteration calculation is used because of the uncertainty of the ratio of activity coefficients, Ya, (Ta = YCdTe/YCdYTe ), which is initially set equal to one [6]. Convergence is reached when the calculated values of YCdTe, Yca and YT~ from two consecutive steps for a modified ratio of the activity coefficients are within a specified toler-

Table 1 Interaction coefficients (in calories) in the Cd-Te system [6]

Cd-Te

Liquid species

a,2 = 5832.0-5.86T fit2 = -6783.0+0.392T at3 = 5938.0-3.272T fl~3 = -513+0.1488T a2~ = 734.0-0.2346T ]~23 = 2.0-0.5577T AH~' = 19169 AS~ ~ = 2.53

Cd( 1 ), Te(2), CdTe(3)

ance (10 2). The converged ratio of the activity coefficients turn out to be about 0.96. The calculated results are shown in the next section.

Having given the species Cd, Te and CdTe in the associated liquid phase, the thermodynamic characterization is completed by assuming an equation for the Gibbs energy of mixing as

3 3

AG= E (15) j=. i=,

where i , j = 1,2,3; 1 represents Cd, 2 - Te, and 3 - CdTe, ai i =a/ i , a i i= 0 and ~ii = - ~ii, AGo is the Gibbs energy of dissolution of CdTe. The a i i, /3i; are composition-independent interaction coefficients, and form the adjustable parameters of the model along with AG °. Table 1 lists these interaction parameters which are determined by a simultaneous fit to the diverse phase diagram and thermodynamic data for the Cd-Te binary system [6,4]. In the table, AH~ ~ and AS ° 3 are the enthalpy and entropy of dissociation of CdTe, with the relation AG°3 = A H °

- TAS °. It can be seen that these interaction coefficients are linear functions of temperature. The relative chemical potential (or partial molar free energy) due to mixing is defined as

= ( 1 6 )

using Eq. (15) for AG m, we may write

3

~,(species) = 2 Y'~ [aip + ( Y i / 2 - Y,)]3,i] Y; i - I

3 3

- E E (ai i + 2~i iYi)Yi~ i=1 j = l

- 6( p , 3 ) A G °, (17)

528 X. lie et al. / Journal of Crystal Growth 171 (1997) 525-530

where i,j,p = 1,2,3, respectively. The relative chemical potentials after mixing are then equal to the species chemical potentials before mixing plus the quantities due to mixing,

~j(species) = RgT In ~ + ~;(species) . (18)

The total enthalpy of mixing is defined as

A Hm(species) = ~'hi(species), (19)

and

A Hm(gr. at. component)

= (1 + Y3) IAHm(species), (20)

where ~i is the relative partial molar enthalpies of species,

hi(species) = ~(-~JT) , (21) a ( 1 / r ) Y,,~2,Y,,,'

hj(species) is a constant if the interaction coefficients are linear functions of temperature. The relative heat capacity at constant pressure is given by

ACp(mole species) = -~]N,.N:.P (22)

and

ACp(gr. at. components)

~ , t Oln Y3X

(23)

The heat capacity of the solution may be expressed as

Cp(gr. at. components)

= ACp(gr. at. components) + xcdCp,ca

+ XTe Cp,Te, (24)

where Cp,cd and Cp,Te are the heat capacities of the pure liquid elements, Cd and Te respectively at 750°C.

3. Results and discussion

On the basis of the thermodynamic model given in Section 2, the thermochemical properties of the

0.35

~. 030

0.25

0.20

g 0 . 1 5

0 . 1 0

0.05

- - - - . . . . . X(Te)=0.76

0.81

0.85

0.89

0.92

0.95

O . O C , r , , i L , , , I , , , , I , , , , , , ,

500 600 700 800 900 1000

Temperature, (°C)

Fig. 2. The mole fraction of the CdTe liquid species under different atomic compositions of Te and temperatures.

solution are obtained using the relevant thermodynamic equations. The temperature dependence of mole fractions of liquid species can be calculated from Eqs. (3), (4) and (14). Fig. 2 shows the calculated CdTe mole fractions as a function of solution temperature for different atomic compositions. The CdTe mole fraction slightly decreases with increasing temperature at an atomic composition of Te, indicating the weak temperature dependence. The decrease in mole fraction of CdTe (YCdVe) is due to increased thermal energy of the CdTe molecule for its dissolution with temperature increase. This weak temperature effect on CdTe dissociation is probably because the CdTe species possess higher ionic contribution to the bond energy, which results in stronger packing of the lattice volume [1,7]. As the CdTe dissociation increases with temperature, the Cd and Te mole fractions increase. It is worth noting that the amount of Cd solute present in the solution is very small since the sum of the mole fractions of the three species is equal to 1. With increasing atomic composition of Te, the CdTe mole fraction decreases as expected. It can be seen that a higher temperature effect on CdTe dissociation is related to a less dilute solution with a lower Te composition. The different compositions give rise to differences in liquid solution properties which affect the chemical bonding state and the dissociation of the CdTe compound. The amount of Cd species seems important, influenc- ing the CdTe equilibrium state since the very high Te composition (X(Te)=0 .95) gives rise to the negligible effect of temperature on the dissociation of CdTe.

x. Ye et al. / Journal of Cr).'stal Growth 171 (1997) 525-530 529

The heat capacity and the total enthalpy of mixing in the liquid of certain composition can be calculated from Eqs. (15) to (24) using the interaction coefficients listed in Table 1. Fig. 3 shows the calculated liquid heat capacity at different compositions (X(Te)). The heat capacity is more sensitive to temperature change at lower temperatures. With rise of temperature, both atomic vibration and interactions of the atoms and species in the liquid are greatly enhanced at lower temperatures and the temperature gradients of the potential and kinetic energies are high. Since heat capacity is the internal energy change with respect to temperature, which involves change in the potential and kinetic energies, it largely increases at lower temperatures. At high temperatures, the potential energy decreases due to less interaction of the species because of increase in entropy. The increase in the kinetic energy therefore compensates for the loss of potential energy and an energy balance is reached. Therefore, the heat capacity becomes constant over a high temperature range. The heat capacity is influenced by the solution composition. A non-dilute solution (e.g. for X(Te)= 0.76) has higher concentrations of unlike atoms and species introducing higher degree of interaction and potential energy. Therefore, the heat capacity is larger. The temperature sensitivity of heat capacity at lower temperatures leads to unstable solution, detri- mental to crystal growth. For this reason, the liquid solution for CdTe crystal growth by the traveling heater method may be chosen at the Te composition of 0.8 and growth temperature of 800°C.

10.6

10.4

E ~ 1 0 . 2

~ 1 0 0

~ 9.6

o _ 9 . 6

j.

/

X(Te)=0.76 _ J-

0.81 I - . . . . .

0.85

~ - J J - ~ 0.89

3~ 9.4 - - - ~

9 2

9 , 0 i i ~ i i i i i i i i J i i

500 600 700 800 900 1000

Temperature, (%)

Fig. 3. The heat capacity in the Te-rich solution under different atomic compositions and temperatures.

-1500

-2000 g o

# 25o0

I -3000

-3500

I -4000

-4500

X(Te)=0.89 . . . . . .

0,85 . . . .

0.81 .

0,76 - " - - -

500 600 700 800 900 1000

Temperature, (~C)

Fig. 4. The enthalpy of mixing in the Te-rich solution under different atomic compositions and temperatures.

It is noted that the decomposition of CdTe with increasing temperature, though at nearly constant rate (see Fig. 2), is caused by different contributions from atomic vibration and species interactions. The lower temperature decomposition is probably due to the actions of more atomic interaction and some atomic vibration, while the atomic vibration makes the major contribution to the higher temperature decomposition.

Fig. 4 shows the calculated liquid enthalpy of mixing for different Te compositions. Since the enthalpy of mixing is the difference between the enthalpies after and before the mixing, the negative values of the enthalpy indicates that the mixing process is an endothermic process. It is shown that the enthalpy of mixing decreases with increasing temperature. This is because high temperature solution involves high entropy and more open spaces for liquid mixing. Also shown is that more heat is evolved at lower Te compositions. This suggests that non-dilute solutions mix less easily.

4. Concluding remarks

Temperature dependences of associated liquid species, heat capacity, enthalpy of mixing in the Te-rich liquid for CdTe crystal growth have been calculated. The calculation is carried out by the solution thermodynamic analysis using the full associated liquid model. It has been shown that the liquid consists of a high concentration of CdTe associates

530 X. Ye et al. /Journal of C~stal Growth 171 (1997) 525-530

with small amounts Cd species. This may provide a basis for a simplified mass transport analysis consid- ering the main solute only. The CdTe associates have strong chemical interaction between Cd and Te in the liquid state and increase in temperature does not significantly affect CdTe dissociation over the temperature range of interest. A preheating process to dissociate CdTe in the liquid solution has to take this factor into account. The liquid mixing process is an endothermic process and the mixing enthalpy is quite large. The heat capacity of the liquid at lower temperatures has larger temperature sensitivity. A solution of stable heat capacity may be achieved at Te composition of 0.8 and growth temperature of 800°C. Calculations provide information on establishing growth conditions. Further efforts on developing preheating process and growth kinetics analysis are required for high quality CdTe crystal growth.

References

[1] P. Rudolph, H.J. Koh, N. Schafer and T. Fukuda, private communication, 1995.

[2] P. Rudolph, K. Umetsu, H.J. Koh and T. Fukuda, J. Jpn. Assoc. Crystal Growth 21 (1994) 166.

[3] A.S. Jordan, Met. Trans. 1 (1970) 239. [4] C.-H. Su, J. Crystal Growth 78 (1986) 51. [5] X. Ye, PhD Dissertation, Department of Mechanical Engineer-

ing, University of Victoria, BC, Canada, 1995. [6] R.F. Brebrick, C.-H. Su and P.-K. Liao, in: Semiconductors

and Semimetals, Vol. 19, Eds. R.K. Willardson and A.C. Beer (Academic Press, New York, 1983).

[7] V.M. Glazov, S.N. Chizhevskaya and N.N. Glagoleva, in: Liquid Semiconductors, Soviet Physics-Semiconductor (Plenum, New York, 1969).

[8] J.C. Brice, Progr. Crystal Growth Characterization 13 (1986) 39.