Thermodynamic states and phase diagrams for bulk-incoherent, bulk-coherent, and epitaxially-coherent semiconductor alloys: Application to cubic (Ga,In)N

The morphology and microstructure of $A_{1-x}{B}_{x}C$ semiconductor alloys depend on the type of thermodynamic states established during growth. We distinguish three main cases: (i) bulk-incoherent structures occur when the alloy grows without being coherent with an underlying substrate and when each of the possible alloy species—phase separated $AC$ and $BC$ constituents, random $A_{1-x}{B}_{x}C$ alloy, or ordered ${\left(AC\right)}_n/{\left(BC\right)}_m$ structures—maintain their own lattice structures and lattice constants, giving up mutual coherence. Bulk incoherence is common in thick films with sufficient dislocations. For cubic (Ga,In)N, bulk-incoherent structures are found to have a positive excess enthalpy $\Delta H_{\text{bulk}}^{\text{incoh}}>0$ and, thus, to phase separate. (ii) Bulk-coherent structures occur when the alloy grows without being coherent with a substrate, but each of the possible species internal to the alloy film is forced to be coherent with the film matrix. Thus, the constituents $AC$-rich and $BC$-rich solid solution phases share the same lattice structure at their interface, leading to internal strain that destabilizes the $AC+BC$ separated constituents. This can expose the intermediate ${\left(AC\right)}_n/{\left(BC\right)}_m$ ordered phases as stable structures with respect to the strained constituents, i.e., $\Delta H_{\text{bulk}}^{\text{coh}}<0$. Bulk coherence is applicable to growth when the development of dislocations is inhibited, e.g., small size precipitates in the alloy matrix. For cubic (Ga,In)N alloy, we find that the coherent ground state phases are three ordered superlattice structures: ${\left(\text{InN}\right)}_2/{\left(\text{GaN}\right)}_2$ $\left(=\text{chacolpyrite}\right)$, ${\left(\text{InN}\right)}_3/{\left(\text{GaN}\right)}_1$, and ${\left(\text{InN}\right)}_4/{\left(\text{GaN}\right)}_1$, along (201) [and its cubic symmetry equivalent, i.e., (102), (210), etc.] crystal direction. (iii) Epitaxially coherent structures occur when the alloy is made coherent with an underlying substrate, e.g., in thin film pseudomorphic growth. Depending on the substrate, the formation enthalpy $\Delta H^{\text{epi}}<0$. For cubic (Ga,In)N grown on GaN (001) substrate, we find that the stablest epitaxial phases are chalcopyrite and the ${\left(\text{InN}\right)}_4/{\left(\text{GaN}\right)}_1$ superlattice along the (210) crystal direction. Here, we calculate, from first principles, the formation enthalpies of cubic zinc blende (Ga,In)N alloy under the three forms of thermodynamic states indicated above to establish a cluster expansion, from which we calculate the finite-temperature phase diagrams. This illustrates how the thermodynamic constraints during growth can radically alter the alloy phase behavior and its microstructures.

Figure 1

Schematic plot of energetic orders in bulk-incoherent and bulk-coherent alloys. Because of the lattice mismatch between $AC$ and $BC$, the ordered structure $\sigma$ has a higher energy than that of incoherent phase separation $AC\left(a_{AC}\right)+BC\left(a_{BC}\right)$, i.e., $\Delta E\left(\sigma \right)>0$. In the bulk-coherent state, the strain energy cost $\Delta E_{CS}^{\text{min}}\left(x\right)$ to maintain the lattice coherence for $AC$-rich and $BC$-rich solid solutions at the interface increases the coherent phase separation energy and, thus, the formation enthalpy $\Delta H_{\text{bulk}}^{\text{coh}}\left(\sigma \right)$ in the bulk-coherent state becomes negative.

Figure 2

Schematic plot of energetic orders in bulk-incoherent and substrate-coherent epitaxial alloys. Because of the lattice mismatch between $AC$ and $BC$, the ordered structure $\sigma$ has a higher energy than that of incoherent phase separation $AC\left(a_{AC}\right)+BC\left(a_{BC}\right)$, i.e., $\Delta E\left(\sigma \right)>0$. In the presence of the substrate, the strain energy $U^{\text{epi}}\left(\sigma ,a_{\text{sub}}\right)$ destabilizes the phase separation $AC\left(a_{\text{sub}}\right)+BC\left(a_{\text{sub}}\right)$ and then the formation enthalpy $\Delta H^{\text{epi}}\left(\sigma \right)$ becomes negative.

Figure 3

(Color online) Formation enthalpies and phase diagrams of zinc blende (Ga,In)N semiconductor alloy under three thermodynamic states: (a, d) bulk-incoherent thermodynamics, (b, e) bulk-coherent thermodynamics and (c, f) substrate-coherent-epitaxial thermodynamics on (001) GaN substrate. In bulk-coherent alloys (b), three (201) superlattice (SL) structure chalcopyrite(CH) ${\left(\text{InN}\right)}_2/{\left(\text{GaN}\right)}_2$, famatinite ${\left(\text{InN}\right)}_3/{\left(\text{GaN}\right)}_1$ and ${\left(\text{InN}\right)}_4/{\left(\text{GaN}\right)}_1$ are predicted to the ground states. For substrate-coherent-epitaxial alloy, two (210) superlattices chalcopyrite ${\left(\text{InN}\right)}_2/{\left(\text{GaN}\right)}_2$ and ${\left(\text{InN}\right)}_4/{\left(\text{GaN}\right)}_1$ are predicted to be ground state structures.