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Mat. Res. Soc. Symp. Proc. Vol. 512 ©1998 Materials Research Society

structure is estimated to be 6.66 kcal/mol, which is almost the same value as that of InGaN with the zinc-blende strcuture. This suggests that an unstable mixing region exists for a wide In content region for both the wurtzite and the zinc-blende structures. The unstable mixing region of InGaN with a wurtzite structure is wider than zinc-blende InGaN. The critical temperatures for phase separation in InGaN are estimated to be 1967 K and 1668 K for the wurtzite and zincblende structures, respectively. In order to show the validity of our calculated results, we compare calculated experimental results for the InGaAs system. The calculated results agrees well with the experimental results. THE VALANCE FORCE FIELD MODEL IN WURTZITE STRUCTURE The ideal wurtzite structure is composed of two identical hexagonal perfect close-packed sublattices, where c/a = (8/3)"/2, shifted along the c axis by u = 3/8, as shown in Fig. 1. Each atom is surrounded by four atoms of the other sublattice. In the real case, ideal how ever, the structure deviates from the (8/3)"/2. case, that is, c/a is usually lower than Table I gives the a, c, and u values in GaN and InN [10]. Thus, the structure is distorted along the c axis in both cases. Taking into consideration the distortion in GaN and InN along the c axis, the distance, d', for the bond between atoms with the same x and y coordinates (atom j and atom 4 in Fig. 1) and the equatorial distances, d', for the bonds between atoms having at least different x or y coordinates (atom j and atoms i =1 -3 on Fig. 1), are given as follows,

d= d.4 = uc2 de-

0 [0100]

....................... [1000] ................. .................

d

d 4

C .........

......................................

............

.

....

Fig. I Wurtzite structure. For a given atom j, one may distinguish bonds with atoms having the same x and y coordinates (atoms labeled 4) and atoms having at least different x or y coordinates (atoms 1~3).

(1) (2)

[a /3 + (1/2

0e

1

In the same way, angles between the bonds are not strictly equal to arcos(-1/3) = 109.47 '. There are two types of angles in wurtzite structures, one is the angle around j atoms involving neighbors at different z values, 0, and the other is the angle with neighbors at the same z coordinates, 0,. 0, and 0. are given as follows, 0== 0Ci4 = arcos[(u -

0" =e,ijk =arcos[1

-

(3)

1/2) c/de], a2 /2dc]. (k = 1-3)

(4)

Table. 1: Lattice parameters (a, c) and sublattice translation parameter (u) a(A) c(A) c/a u

GaN 3.182 5.189 1.631 0.377

InN 3.540 5.705 1.612 0.378

Ideal wurtzite 1.633 0.375

In the case of a ternary system which consists of two different binary compounds, such as InGaN, the equilibrium bond length between Ga-N and In-N are different from each other so that each atom moves to reduce the strain energy. In this case, the interatomic distances given by the Vegard's Law and the natural interatomic distances do not agree. In other words, lattice relaxation occurs. 292

In the