Liquidus calculations for III-N semiconductors

CALPHAD Printed

Vo1.8, No.4, in the USA.

pp. 343-354,

LIQUIDUS

1984

0364-5916/84 $3.00 + .OO (cl 1984 Pergamon Press Ltd.

CALCULATIONS

FOR III-N

SEMICONDUCTORS

R. D. Jones and K. Rose Center for Integrated Electronics Rensselaer Polytechnic Institute 12181 U.S.A. Troy, New York

ABSTRACT

:

Liquidus curves for III-N semiconducting compounds (AlN, GaN, and InN) are calculated. A quasi-chemical equilibrium approach also known as the first approximation in regular solution theory is These calculations are based used to calculate the T-X data. primarily on the heat and entropy of formation and the heat and A semi-empirical method is used to estimate entropy of fusion. the entropy and temperature of fusion values of the nitrides. The entropy of fusion of AlN is estimated to be 15.2 eu/mole with a melting point of 2800°C; for GaN, ASF = 16.1 eu/mole with T, = 17OO'C; and for InN, AS F = 14.5 eu/mole with T, = 1200°C. These liquidus calculations are used to compute pressure of N2 over both the column III rich and nitrogen rich regions of the nitrides. Implications for crystal growth are discussed. 1.

Introduction

The selection of an optimum crystal growth process such as growth from the melt, growth from solution, or growth from the vapor is based primarily on From the information derived from phase diagrams of the material to be grown. liquidus curve of the compound and its corresponding pressure-temperature curve, growth parameters such as the temperature range required for heating, the type of container compatible with this temperature range, and the necesIn the case of growth from the sary growth atmosphere can be determined. melt, a steep liquidus curve as one moves away from the stoichiometric A small composition indicates the degree of temperature control required. deviation of the melt from the composition of the solid will require a major readjustment of the growth temperature. As temperature-composition curves do not exist for the semiconducting nitrides: AlN, GaN, and InN, the purpose of this paper is to calculate these The importance that such calculations have played in the development curves. of GaAs technology is well pointed out by Joyce (1). The semiconducting nitride compounds have received only a small amount of attention, relative This is primarily to the more conventional III-V compounds such as GaAs. (especially in the case of GaN and InN) due to the high vapor pressures at This has led to difficulties in growing their respective melting points. large single crystals and also in obtaining p-type material for device fabriThis latter difficulty probably stems from the presence of cation (2,3,4). nitrogen vacancies which are linked to the high vapor pressures. 2.

The Entropy

of Fusion

Equation

A well known relationship developed by Vieland of a binary compound AB is the following: _______---___-_______-___________-______ Received

16

March

1984

343

(5) for the liquidus

curve

R.D. JONES and K. ROSE

344

sl

sl

1

'A 'B -(2.1) [ 'A 'B Sl Sl are the activity coefficients for a stoichiometric liquid, X , yB where Y is the Bole fraction of the column V element in the composition, ASF is the entropy of fusion per mole of compound assumed to be independent of temperature. It should be noted that in (2.1) ASF has the dimensions of eu/mole and not eu/g. atom as used by Vieland. TM is the melting point of the stoichiometric compound. ln

This relationship will be referred to as the entropy of fusion equation. Given data on the activity coefficients one can then calculate a liquidus curve for the compound. However, since experimental solubility data do not exist, it will be necessary to calculate the activity coefficients as well. This is done by developing a second T-X equation based on the heat of formation rather than on the entropy of fusion (6). 3.

The Heat of Formation Equation

A T-X equation relying on the heat of formation is derived based on a chemical potential approach (6). Consider a compound AB. When the liquid phase is in equilibrium with the solid phase, the chemical potentials of the composition can be equated. S 1 uAB = 'AB

(3.1)

where

1,1+ uAB 'A

1 u1B

(3.2)

and

S s+ uAB = 'A

s uB

(3.3)

The chemical potential of liquid component A (or B) is related to the chemical potential of pure liquid A (or B) by 1

pi

=

l,o ui

+ RTlna

(3.4)

i

where i is the component A (or B), and a. is the activity of A (or B) in the melt. The superscript '0' indicates a p&re liquid. From this it can be shown that RTlnaA + RTlnaB = (us A

(3.5)

The right hand side of this equation is just the molar free energy of formation, AGf. Then AGf

= RTlnaA

+

RTlnaB

(3.6)

or in terms of the activity coefficient y, since a = yx *Gf = RTln [yA(l - X)] + RTln [y,Xl Using Vieland's equation (2.1), Eq. (3.7) can be written as AGf

=

RTln

sl sl 'A 'B [ 4

(3.7)

1

- ASF (TM - T)

(3.8)

From the appropriate temperature derivatives, the enthalpy and entropy of formation are 1 sl yB

--++-

ASf

=

- RT

--1 sl YA

ayz' aT

aYzl 3T

1 -

AH

F

(3.9)

+ $

$I-

Rln[%]+

ASF

(3.10)

LIQUIOUS

CALCULATIONS

FOR III-N SEMICONDUCTORS

The usefulness of Eq. in the next section.

where AHF is the heat of fusion. the liquidus curve will be shown

4.

345

(3.9) in determining

The QCE Model

To evaluate the activity coefficients appearing in Eq. (2.1) and Eq. equilibrium (QCE) theory of Guggenheim is used (7). (3.9), the quasi-chemical QCE theory is used because of its reasonable assumption of a non-random The four basic distribution of atoms due to the various bonding arrangements. assumptions of this theory are (8) 1.

Each

2.

Nearest

atom

has z nearest

3.

Only the configurational free energy of mixing tional free energy of mixing is neglected).

4.

The distribution expression:

neighbors

N

neighbors.

interact

of atoms

AANBB=

1

N;B

a

pairwise

with

is calculated

e

an interchange

using

energy

is considered a mass

action

~1.

(vibratype of

2C1/ZRT

where 2NAA

+

NAB

=

2NBB

+

NAB

=

ZNA ZNB

an inIf one starts with two pure crystals A and B, and interchanges terior A atom with an interior B atom, the total increase in energy is 2ct, all molecules being assumed at rest on their lattice points (7). This increase in energy per molecule is referred to as the interchange energy a which appears in assumptions (2) and (4) above. The activity

coefficients

(7) YA

L

i+S2x 1-X) (l+B)

=

and

‘B 1 p;: where

::

according

can be expressed

1

as

z/2

;;I

(4.1)

z'2

2

1 + 4X(1-X)(n

B

to the QCE model

-1)

a,ZRT

and

n=e

Substituting Eqs. (4.1) and (4.2) into Eq. (2.1) yields Vieland's entropy of fusion liquidus equation in the QCE model. Z was set equal to 6 based on experimental results for InSb (9).. 1+0 61n

1

[ 1 + n - ln[4X(l-X)]

Likewise, AH~

=

the heat of formation

=

- AHF

(4.3) equation,

Eq.

(3.9), is

now

a

+ l+e

(4.4)

e'6RT

The liquidus of the compound is calculated primarily from Eq. depends on three parameters: a(through 6 and n), ASF, and TM.

(4.3) which

;he interchange energy c1 is determined from Eq. (4.4) which also depends on AS and T as well as on AHf. This last parameter is well reported in the literature, flowever, the entropy of fusion and the melting point will have to be estimated.

R.D. JONES and K. ROSE

346

5.

Entropy of Fusion

The entropy of fusion was first estimated using a method suggested by Marina (10) which relies on combinations of column IV elements comparable in molecular weight to the III-V compound. Thus AS;II_V

=

AS&,

+

AS&,

+ Rln4

(5.1)

where Rln4 is due to the entropy of mixing which is defined as (7) Smixing

= =

- R(l-X)ln(l-X) - RXlnX

(5.2)

Rln4 eu/mole for X = 0.5

(5.3)

These values are tabulated in column 3 of Table 1. TABLE 1 Entropy of Fusion Data

Compound

Column IV Combination

Calculated* ASF (eu/mole)

Adjusted ASF (eu/mole)

Van Vechten(l2) ASF (eu/mole)

AlN

C+Si+Rln4

16.0

15.2

16.61

GaN

C+Ge+Rln4

16.1

16.1

16.01

InN

C+Sn+Rln4

14.1

14.5

10.19+

Column IV ASF values from ref. tll).( + probably in error, see discussion below, To obtain a more reasonable estimate, the entropy of fusion values for other III-V compounds were calculated in the same manner and compared with their experimental values. Fig. 1 shows how the experimental values ('13)of ASF for III-V compounds differ from the calculated values. All of the calculated values fall within 10%. It is interesting to note the linear relationship between ASF and AsF . Plotting calculated values on these lines gives empirically adj@?ed val&$'for ASF of the nitrides. These values are listed in Table 1. *

These ASF values are compared with those of Van Vechten's which are based on a quantum dielectric two band theory of electronegativity (12). GaN compares well and AlN is reasonably close (within - 8%). Van Vechten's estimate for InN is probably too low as he points out that there is an error in his calculated optical spectrum for InN. Temperature of Fusion 26 There is a great deal of uncertainty in the literature concerning the melting points of the nitrides. To reduce this uncertainty, the melting points (14) of the III-V compounds were plotted versus their molecular weight. A fairly linear fit (see Figure 2) was found to exist between compounds with a common column III element. From Fig. 2, the following reported values for TM were selected based on their proximity to the linear curves shown: 30730K for AlN (16); 1973'K for GaN (15); and 1473'K for InN (11). Discrepancies in TM could be due to the presence of impurities in the semiconductor which will usually lower the melting point. Another important factor is that sublimation for the nitrides occurs below the melting point at 1 atm. 7. Liquidus Calculation With these estimates and the heat of formation (see Table 2), Eq. (4.4) is used to determine CL. These values are listed in Table 2 and along with ASF and TM are used in Eq. (4.3) to plot the liquidus curves for the III-N compounds. These curves are shown in Figs. 3, 4, 5 along with the melting points for the constituents of each compound. The diagrams are reasonable compared to the other III-V compounds (1, 13, 14 18). Unfortunately,

LIQUIDUS CALCULATIONS FOR

347

III-NSEMICONDUCTORS

19

17

16

15 Inl

14

0 experimental

InSb 13

A predicted

12 ll 11

I

I

I

12

13

I

1

14

15

16

MPEMMENI'AL ASF

I

I

17

16

1

1

19

20

21

, cu/mole

FIO. 1 A ~omparieonof experimentaland cslculatedvalues of ASF for III-V compounde.

3000

‘\ k,AlP

2500

J-i 2ooa . r? 1500

1000

5cQ

I

Ito

60

I

80

I

1

I

loo

120

140

1

160

I

180

1

200

I

220

240

molecular weight FIO. 2 Melting points of the III-V compound8(~8m iunction of mleculsr weight. AlN: x Ref. A Ref.(16)1 CaN: x Ref.(17), A Ref.(15)j InN: x Ref.(l5), A

R.D. JONES and K. ROSE

348

AU(s) + Nab)

I I 1 ' F 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-2501 0

1750

15GC 125C

f 0" . h

1ooC

75C Calf(s)

+

Ga(1)

25C

G&f(s) + Ga(s) I

0

0.1

1

0.2

e

I

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 %'

mole fraction FIG. 4

LIQUIDUS CALCULATIONS FOR III-N

1500

,

I

11 I, 0.1 0.2 0.3 0.4 0.5

,

I

I

I

0.6

0.7

0.8

0.9

I

,

I

I

SEMICONDUCTORS

I

I

I

1250 -

O-

InN(s) + In(s)

-250

0

$1

mle

1.0

fraction

FIG. 5

2

I

I

I

I

,

I

I

I

0.7

0.8

,

1

0

;

-1

-2

-3

-4

,

I

1

1

0.1 0.2 0.3

t

0.4 XN'

I

0.5

1

0.6

I

I

I

0.9

mole fraction

FIG. 6 Activitycoefficientof N2 for AM, Cd?, and InN.

0

349

R.D. JONES and K. ROSE

350

experimental

data

are not available

for direct 2

TABLE Thermodynamic

comparison.

Properties

of AlN,

ASF

GaN,

InN

Compound

(11) AHf Kcal/mole

eu/mole

TM OK

CY Kcal/mole

AlN

-76.1

15.2

3073

-29.4

GaN

-26.2

16.1

1973

-5.4

InN

-33.0

14.5

1473

-12.1

8.

Activity

Coefficients

It is useful to examine the activity coefficients to see how the nitride A plot of yN versus XN indicates that all solutions depart from ideality. three nitrides display a negative deviation from Raoultian behaviour (YN = l), see Fig. 6. GaN appears to be more ideal than either AlN or InN. 9.

Vapor

Pressure

To determine a growth process for the III-N compounds it is necessary to use vapor pressure curves (P-T) in conjunction with the liquidus curves. Using a = yX and a = [P(N )/P"(N )11i2, P-T curves are calculated for AlN, GaN, PofN2) is the vapor pressure of N2 over liquid and InN (see Figs. 7, 8, 6). Its temperature dependence is (19) N2. Log PO(N2) where

P"(N2) Po(N2)

=

3.6138

- 2;z;6;0

(9.1)

.

is in atmospheres. has no clear

physical

meaning

above

the critical

point

(Tc

=

126.3"K, PC 33.5 atm). purposes, a plot of Log = However, for calculation beyond the critical point, since the plot P"(N2) vs. ~-1 can be extrapolated is essentially a straight line (26). An examination of the available vapor pressure date (see Figs. 7, 8, 9) brings out the rather large disagreements that exist for both GaN and InN. When vapor pressure measurements are made, they become more subject to experimental problems as the temperature is increased. These problems are in the nature of errors in the evaporation temperature, excessive residual gas pressure, and reaction of the condensed evaporant phase with the container. As a consequence, the accuracy of vapor pressure data is limited, and a judicious choice is therefore necessary, particularly if the data are relatively old(27). It should be noted that the calculated curves fall squarely in the and middle of the experimental data, adding validity to their calculation also suggesting their usefulness as a reasonable alternative to the widely varying pressure data. A pressure-temperature relation incorporating both the column III rich and nitrogen rich regions of the compounds is also Note the expected high vapor calculated for each nitride (see Fig. 10). Expressions for the calculated pressure along the nitrogen rich liquidus. curves shown in Figs. 7, 8, 9 are given below. AlN:

where

Log P(N2)

=

16.89

-

46,777

(9.2) (9.3) (9.4)

GaN:

Log P(N2)

=

10.72

-

16T870 +

InN:

Log P(N2)

=

12.75

-

16,615 T

P(N2)

is in atmospheres.

LIQUIDUS CALCULATIONS FOR III-N

SEMICONDUCTORS

ld

10*

ld

loo

10-l

10

-2

10-3

10

-4

104/TK-l )

FIG.7 vapor

of

pressure

calculated,_.-Ref.(15),

AlIT._

--Ref.(20).

104

12 10*

101

loo

10-l

1O-2

10-3 4

5

6

7

a

9

10

,04/T , K-l FIG. 8 Vapor pressure of GaN._._Ref.(21),-----ReP.(n).

cs.lculated,_

-Ref.

(15)

351

352

R.D. JONES and K. ROSE

T, 1200

OC

1000

6cx3

800

10*

10-l E’ f _ loo

T

8

7

6

104/T , FIG. vapor pressure -._._._Ref.(23)

ILL

10

9

I.2

K-l

9

of Inri. calculated, ,-Rcf.(24) ,----

-.-Ref. (151, Ref. (25)

N rich

4

2

8

6

10

I2

14

16

104/T , K-l FTC. ylilbr~

*

t

v~I~.me

.

10

of H* along

tile 11qtiauscurve*

LIQUIDUS CALCULATIONS FOR III-N SEMICONDUCTORS

10.

353

Discussion

If growth from the melt is considered, the liquidus curves indicate that to avoid a melt with an excess of the column III element, it is necessary to maintain an over-pressure of nitrogen. Atomic nitrogen is preferable since it is difficult for molecular nitrogen to react with Al, Ga, or In due to the strong triple bond in N2. The P-T curves (see Fig. 10) show that a nitrogen over-pressure of approximately 50 atm. for AlN, 370 atm. for GaN, and 65 atm. for InN would be required to maintain stoichiometry. Thus growth from the melt is technically quite difficult due to the large over-pressures of nitrogen that must be maintained. For comparison, GaAs has a vapor pressure of 1 atm. at its melting point (1238'C) and can be grown from the melt. However, GaP has a vapor pressure of 38 atm. at its melting point (1468OC) and is more difficult to grow. It has to be grown from a nonstoichiometric melt (a solution of 10 to 20 atomic % P) where the melt temperature is less than 12OO'C and the corresponding vapor pressure is now less than 10m2 atm. (28). The excess Ga is removed chemically using dilute HCl yielding crystal sizes less than 1 cm3 and showing n type behavior. The growth of GaN from a nonstoichiometric melt at a workable overpressure of 10m2 atm. would require a temperature of 108O'C and a solubility of less than 1 atomic percent N. This yields an even larger excess of liquid Ga surrounding the crystal than in the case of Gap. The crystal itself will also probably have a Ga excess leading to n type behavior. This growth method then offers technical problems concerning the removal of the excess Ga and maintaining precise control over the stoichiometry of the crystal. Vapor growth appears to be more viable since one would have more control over the growth process at much lower pressures and temperatures. For typical nitrogen deposition pressures are 10m7 atm. physical vapor deposition, This indicates that the substrate temperature of AlN should be less than 1685'C to avoid decomposition of the nitride. For GaN, the temperature of the substrate should be less than 680°C; and for InN, less than 57O'C. Problems in maintaining adequate atomic nitrogen pressures are responThis results in nitrogen sible for the growth of column III rich compounds. vacancies which are thought to be responsible for the n type behavior of the semiconductor nitrides. Acknowledgement Discussions with Prof. J-X Shi (on leave from the Central-South People's Republic of China) have been very of Mining and Metallurgy,

Institute helpful.

References 1.

B. A. Joyce, Crystal Growth (B. R. Pamplin, Pergamon Press, New York (1975).

2.

B. B. Kosicki

and D. Kahng,

J. Vat.

3.

J. I. Pankove

and S. Bloom,

RCA Review

4.

A. N. Krasiko, A. F. Andreeva, V. A. Tyagai, A. M. Evstigneev, V. Ya Malakhov, Sov. Phys. Semicond. II, 1257 (1977).

5.

L. J. Vieland,

6.

R. F. Brebrick,

7.

E. A. Guggenheim,

8.

G. B. Stringfellow,

9.

R. Buschert, I. G. Geib, Ser. -11 cl), 111 (1956).

Acta Met.

Meta.

11,

Trans.

Mixtures

137

Sci.

& Tech.

2,

163

Chap.

5,

593

5, p. 157,

(1969).

(1975). and

(1963).

2, 1657

(1971).

(Oxford Univ.

J. Electrochem.

ed.),

Press,

Sot. 117,

and K. Lark-Horowitz,

1952).

1301

(1970).

Bull.

Am. Phys.

Sot.

R.D. JONES and K. ROSE

10.

L. I. Marina and A. Ya Nashel'skii, Russ. J. Phys. Chem. 43, 963 (1969).

11.

0. Kubaschewski and C. B. Alcock, Metallurgical Thermochemistry, Pergamon Press, London (1979).

12.

J. A. Van Vechten, Phy. Rev. B 1, 1479 (1973).

13.

G. B. Stringfellow, Mat. Res. Bull. 5, 371 (1971).

14.

M. B. Panish and M. Ilegems, Progress in Solid State Chemistry, Vol. 7 (H. Reiss and J. 0. Mccaldin, e&i.) p. 39, Pergamon Press, Oxford (1973).

15.

J. B. MacChesney, P. M. Bridenbaugh, and P. B. O'Connor, Mat. Res. Bull. 2, 783 (1970).

16.

W. Class, NASA-CR-1171

17.

A. G. Fischer, Solid St. Electron. 2, 232 (1961).

18.

R. N. Hall, J. Electrochem. Sot. 110, 385 (19631.

19.

N. A. Lange, Lange's Handbook of Chemistry, 12 ed. (J. A. Dean, ea.) Sect. 10, P. 28, McGraw-Hill, New York (1979).

20.

G. A. Slack and T. F. McNelly, J. Crys. Growth 34, 263 (1976).

21.

R. J. Sime and J. L. Margrave, J. Phy. Chem. 60, 810 (1956).

22.

M. R. Lorenz and. B. B. Binkowski, J. Electrochem. Sot. 109, 24 (1962).

23.

A. M. Vorob'ev, G. V. Evseeva, and L. V. Zenkevich, Russ. J. Phy. Chem. -45 (101, 1501 (1971).

24.

A, M. Vorob'ev, G. V. Evseeva, and L. V. Zenkevich, Russ. J. Phy. Chem. -47 (111, 1616 (1973).

25.

s.

26.

J. H. Hildebrand and R. L. Scott, Solubility of Non-Electrolytes, Rheinhold Publishing Corp., New York (1964).

27.

R. Glang, Handbook of Thin Film Technology (L. I. Maissel and R. Glang, eds.) chap. 1, p. 3, McGraw-Hill, New York (1970).

28.

J. w. Nielsen and R. R. Monchamp, Phase Diagrams: Materials Science and Technology, Vol. 6-111 (A. M. Alper, ea.) chap. 1, p. 28, Academic Press, New York (19701.

(1968).

P. Gordienko, Russ. J. Phy. Chem. -51 (2), 315 (1977).