Theoretical Investigation of Extended Defects in Group-III Nitrides
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... ABCABCABC... perpendicular to the [111] direction. In this notation, each letter stands for an ordered pair of cation and anion layers (we refer to this ordered-pair as simply a layer) and a particular letter refers to the atoms being placed at one of the three possible positions within the basal or (111) plane. In this regard, the wurtzite structure is distinguished by a stacking sequence with repeat period two while the zinc-blende structure has repeat period three. A basal-plane stacking fault, whether in wurtzite or zinc blende, is simply a disruption in the normal stacking sequence. In the wurtzite structure, there are three types of faults denoted I1, 12 and E. 10 The II (intrinsic) fault has stacking sequence ... ABABCBCB... with a single unit of sphalerite stacking inserted into the wurtzite sequence. The 12 (intrinsic) fault has stacking sequence ... ABABCACA... containing two units of sphalerite stacking, and the E (extrinsic) fault has stacking sequence ... ABABCABAB... containing three sphalerite units. A stacking-fault energy is defined as the energy difference between faulted and unfaulted structures. This difference is given in terms of either a unit area in the basal plane, or the area of a primitive unit cell in which case it is referred to as a reduced stacking-fault energy. In this study, stacking-fault energies were calculated using an approach employed previously by Cheng et al.I I to study polymorphism in SiC. In this approach, the energy of a periodic sequence of N layers is cast in the form of a 1-dimensional Ising-type model
NE = NEo -
_Jni
(1)
i,n
Here, E0 is a reference energy, Jn is an interaction energy between n"' neighbor layers, and the summations are over all N layers. The spin parameters, Y, are defined in terms of the ordering between adjacent layers. We assigned AB, BC and CA pairs a value of +1 and BA, CB and AC pairs a value of- I with assignments made moving from left to right through a stacking sequence. Interactions beyond the third-neighbor layers have been neglected yielding 1, 12 and E model energies given by the expressions (-2J, + V2 - 6J 3), (-4J1 + 4J 2 - 4J 3) and (-6J 1 + 4J 2 - 6J 3), respectively. The interaction energies were determined from density-functional calculations for the 2H, 3H, 4H and 6H polytypes. In these calculations, lattice constants within the basal plane and separations between adjacent layers were fixed at the relaxed values found for wurtzite. As a result, we use the notation 3H for the unit cell with repeat period three instead of 3C which would denote the fully relaxed cubic structure. Model energies per layer for these polytypes are given by the expressions E21 = Eo E 3H =
+ J,
(2)
J2 + J3
E0 - J- /
(3)
- 3
E4, = Eo + J2 E6H = Eo -
(4)
I JI + 1J2 + 3
3
(5)
J3.
796
Interaction energies were extracted from suitable combinations, substituting density-functional energies on the left-hand sides. General features of the density-functional calculations have been described previously. 12 Convergence of the polytype energies
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