Entropic stabilization and retrograde solubility in Zn${}_4$Sb${}_3$

Zn${}_4$Sb${}_3$ is shown to be entropically stabilized versus decomposition to Zn and ZnSb through the effects of configurational disorder and phonon free energy. Single-phase stability is predicted for a range of compositions and temperatures. Retrograde solubility of Zn is predicted on the two-phase boundary region between Zn${}_4$Sb${}_3$ and Zn. The complex temperature-dependent solubility can be used to explain the variety of nanoparticle formation observed in the system: formation of ZnSb on the Sb-rich side, Zn on the far Zn-rich side, and nano-void formation due to Zn precipitates being reabsorbed at lower temperatures.

Figure 1

Conventional hexagonal unit cell of Zn${}_4$Sb${}_3$. The Sb sublattice (orange) is composed of isolated Sb and Sb dimers. The Zn “A” sublattice (blue) is $90\%$ occupied. The “B” (green), “C” (brown), and “D” (pink) Zn sublattices are $5\%$ occupied. The primitive rhombohedral cell is 1/3 the volume of the conventional cell.

Figure 2

Formation enthalpy at 0 K for Zn${}_4$Sb${}_3$ configurations with respect to Zn and ZnSb. The solid black line represents the convex hull of structures confined to the Zn${}_4$Sb${}_3$ lattice. Supercell calculations in green were done using the conventional unit cell, each composed of three primitive unit cells. The dashed line shows the predicted energy for supercells following the independent cells approximation.

Figure 3

Formation free energy due to phonons with respect to Zn and ZnSb of the lowest energy Zn${}_4$Sb${}_3$ configurations at each composition. There is a more favorable contribution to the formation energy for the more disordered Zn-rich configurations, evidenced by a steeper slope with temperature. Two configurations of Zn${}_{13}$Sb${}_{10}$ are in good agreement to 1 meV per atom.

Figure 4

Single-phase stability region for Zn${}_4$Sb${}_3$. Solid black line corresponds to the charge-balanced composition. Stable compositions to the left of the line will result in a $p$-type semiconductor. (Inset) A close-up of the region of retrograde Zn solubility. The orange band corresponds to compositions estimated from charge carrier measurements.[11]

Figure 5

Possible phase-stability regions of Zn${}_4$Sb${}_3$ and Zn${}_8$Sb${}_7$ under conditions favorable to the formation of Zn${}_8$Sb${}_7$. The dotted line represents the phase boundary of Zn${}_4$Sb${}_3$ in the absence of Zn${}_8$Sb${}_7$.