Effects of the Dislocation Density and Surface Energy on Phase Diagrams of the S-K Growth Mode for the GaInN/GaN and GaP
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CALCULATION METHOD The calculation method is essentially the same as the previously reported one.24 The schematic geometries of a GaInN strained layer on GaN substrate (the FM mode) and a GaInN cluster on a GalnN wetting-layer on GaN substrate (the SK mode) are shown in Fig. 1. Figure 1 illustrates the fundamental structures for the case of two lattice layers thick. The total volume of the cluster is equal to that of the wetting-layer under the cluster as shown in Fig. 1(b). The total volume of the strained layer with two lattice layers thick shown in Fig. l(a) is equal to that of the cluster and wetting-layer shown in Fig. 1(b). Gaa~nN
GalnN 4 4ltielyr
1Ilattice 25 layer
25A
20 lattice layers
20 lattice layers
GaN
GaN
(b) Volmer - Weber mode
(a) Frank - van der Merwe mode
Fig. 1. Schematic geometries of (a) a GaInN film on a GaN substrate (the FM mode) and (b) a GaInN cluster on a GaInN film on a GaN substrate (the SK mode) for the case of two lattice layers. In this calculation, the size of the layer, cluster and substrate are shown in Fig. 1. The size of the substrate is 1o0A x 25A base and 20 lattice layers thick (I 1 OA). The thickness of the strained layer is two lattice layers for the FM mode. The thickness of the wetting-layer is one lattice layer for the SK mode. The shape of the cluster is assumed to be rectangular and the size is 25A long, 25A wide, and four lattice layers thick. The cluster has 1/4 the length of the substrate. For the fundamental structures of n lattice layers thick, the SK mode structure has a cluster with 4 x (n-1) lattice layers thick on a wetting layer with one lattice layer thick on GaN substrate. The thickness of the wetting layer is always one lattice layer for the GaInN/GaN structure. For the GaPSb/GaP structure, the SK mode structure has always a cluster with 4 lattice layers thick on a wetting layer with (n-I) lattice layers thick on GaP substrate. The FM mode structure has a strained layer with n lattice layers thick on the substrate, and the VW mode structure has a cluster with 4 x n lattice layers thick on the substrate. For the calculation of the strain energy, we used the method which was developed on the basis of the elastic model by Nakajima et al.' in order to calculate the precise stress distribution of an island layer on substrate. The total strain energy of each structure G- which includes the strain energy of the layer, cluster, buffer layer and substrate is given by
286
m
Gst = ZU, 1=(
i=1 2Ei where U, is the elastic strain energy in the ith imaginary thin layer, m is total number of imaginary thin layers which constitute each structure, and a, EB,A,, and d, are the stress, Young modulus, surface area and thickness of the ith imaginary thin layer. The strain energy in the FM, SK and VW mode structures with misfit dislocations was calculated on the bases of the assumption that misfit dislocations spreaded all over epitaxial layers and the stress relaxed all over them. On the surface between the vapor and solid phases, the surface energy per unit area
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