Local structural evolution in the anionic solid solution $\mathrm{Zn}{\mathrm{Se}}_x{\mathrm{S}}_{1-x}$

The century-old Vegard‘s law has been remarkably accurate in describing the evolution of the lattice parameters of almost all solid solutions. Contractions or expansions of lattice parameters of such systems depend on the size of the guest atom being smaller or larger than the host atom it replaces to form the solid solution. This has given rise to the concept of “chemical pressure” in analogy to the physical pressure. We have investigated using EXAFS the evolution of the local structure in terms of atom-pair distances extending up to the third-nearest neighbors in the family of compounds, $\mathrm{Zn}{\mathrm{Se}}_x{\mathrm{S}}_{1-x}$ as an example of an anionic solid solution, in contrast to all previous studies focusing on cationic solid solutions. Our results establish several common features between these two types of solid solutions, while strongly suggesting that the concept of a chemical pressure is inaccurate and misleading. Most interestingly, we also find a qualitative difference between the cationic solid solutions, reported earlier, and the anionic solid solution.

Figure 1

(a) X-Ray diffraction (XRD) patterns of $\mathrm{Zn}{\mathrm{Se}}_x{\mathrm{S}}_{1-x}$ solid solution for different values of $x$. The peaks shift systematically towards lower $2\theta$ values with the increase in selenium concentration $\left(x\right)$ as can be seen with the help of the vertical dashed reference line at ${45}^{\circ }$. (b) The lattice parameter $a$ (in Å) from experiment (red-solid circles) and calculations (blue-open circles). The experimental points are obtained from Rietveld analyses of the diffraction patterns as a function of the composition $x$.

Figure 2

$k^2$-weighted EXAFS oscillations obtained at (a) the Zn K-edge, (b) Se K-edge, and (c) S K-edge for different compositions ($x$) of $\mathrm{Zn}{\mathrm{Se}}_x{\mathrm{S}}_{1-x}$. The corresponding Fourier transforms are shown for the Zn K-edge (d), Se K-edge (e), and S K-edge (f). Details of data analysis are given in the Supplemental Material [29].

Figure 3

(a) Experimentally obtained bond distances: Zn–Se (black-solid triangles) and Zn–S (green-solid squares from Zn K-edge and red-solid circles from S K-edge analyses) belonging to the first coordination shell as a function of the composition along with the calculated bond distances shown with corresponding blue-open symbols. (b) Composition averaged nearest-neighbor bond distances (red-solid circles from experimental estimates and blue-open circles from the calculations) as a function of the composition, showing a close adherence of the averaged cation-anion bond distance to the expectation based on the Vegard‘s law.

Figure 4

(a) Experimentally obtained two Zn–Zn distances (long: black-solid triangles, short: red-solid circles) for the second coordination shell as a function of the composition along with the corresponding calculated quantities shown with blue-open symbols. (b) Experimentally obtained Se–Se (black-solid circles), S–S (red-solid triangles), and Se–S (green-solid stars) distances as a function of the composition, along with the calculated quantities with corresponding blue-open symbols.

Figure 5

Zn–Se–Zn (black-solid circles), Se–Zn–Se (red-solid up triangles), S–Zn–S (purple-solid stars), and Zn–S–Zn (magenta-solid down triangles) bond angles obtained from first and second coordination shell bond distances presented in Figs. 3, 4, and 4. S–Zn–S bond angles and Zn–S–Zn bond angles for dilute S compositions ($x\ge 50\%$) could not be estimated with any reasonable accuracy due to the large uncertainties in the S–S and short Zn–Zn bond distances (see Fig. 4). The average angles evaluated from the calculations are also shown with corresponding blue-open symbols, indicating a very good agreement in most cases. The experimentally obtained S–Zn–S angles appear to be slightly shifted with respect to the calculated values, as shown by the dc-shifted blue dashed line; this mismatch is within the uncertainty of estimating these angles from the individual bond distances. All angles obtained for the two pure compounds are in good agreement with the tetrahedral bond angle ($109.{47}^{\circ }$).

Figure 6

Experimental estimates of third nearest-neighbor Zn–Se (black-solid circles) and Zn–S (red-solid triangles) distances as a function of the composition $x$. The same bond distances obtained from calculations are shown with corresponding blue-open symbols.

Figure 7

Relative deviations of the rates of change of various interatomic distances with the composition from those expected from an application of Vegard‘s law plotted as a function of the corresponding interatomic distance, characterizing different coordination shells of the solid.