Energetics of quaternary III-V alloys described by incorporation and clustering of impurities

The energetics of alloy formation is generally modeled either by explicit sampling of the possible alloy configurations or by considering only noninteracting impurities in the dilute limit. We describe a model that bridges the two approaches by taking into account the thermodynamic probability to form small clusters by association of impurities, thereby extending the validity of the impurity model to higher concentrations. Since we express the alloy energetics in terms of pair and cluster binding energies there is no need for computationally intensive sampling over the full configuration space. The application to the ${\text{Ga}}_{1-x}{\text{In}}_x{\text{P}}_{1-y}{\text{N}}_y$ highlights the importance of short-range ordering due to “small atom/large atom” correlation in such quaternary III-V alloys.

Figure 1

(Color online) Example of a model alloy: total concentrations of impurities $A$ or $B$ as a function of their formation enthalpy. $\Delta H_A=\Delta H_B$ for the isolated $A$ and $B$ atoms, and $\Delta H_{AB}=\Delta H_A+\Delta H_B$ $\left(E_{\text{b}}=0\right)$ for the close $AB$ pairs. Triangles (‘‘addition’’): the concentrations are determined according to Eq. (2) and the isolated and paired occurrence of $A$ and $B$ are simply added. Squares (‘‘exact’’): the exact result for this particular model case. Diamonds (“present model”): the result obtained by accounting for overcounting and double-counting effects.

Figure 2

(Color online) Equilibrium solubility of In and N in GaP matrix. Solid symbols correspond to alloy with clustering (with SRO) while open symbols denote random mixing (w/o SRO).

Figure 3

(Color online) Formation enthalpy of ${\text{Ga}}_{1-x}{\text{In}}_x{\text{P}}_{1-y}{\text{N}}_y$ in the lattice matched ratio $x:y=2.12:1$. (a) Random alloy calculated with VFF in large supercells. (b) Present model for $\Delta H$ of the random alloy including statistical formation of clusters (w/o SRO). (c) $\Delta H$ due to noninteracting ${\text{In}}_{\text{Ga}}$ and ${\text{N}}_{\text{P}}$ impurities. (d) Present model for $\Delta H$ of the alloy with SRO at $T=1000\text{K}$.

Figure 4

(Color online) (a) Formation enthalpy of ${\text{GaP}}_{1-x}{\text{N}}_x$ as a function of N concentration. Random alloy calculated with VFF in large supercell (explicit random, solid line). Present model for concentration-dependent $\Delta H$ of noninteracting ${\text{N}}_{\text{P}}$ impurity (open triangles). Present model for concentration-dependent $\Delta H$ of interacting ${\text{N}}_{\text{P}}$ impurity (solid triangles). (b) Variation in the average bond lengths as a function of N concentration.