Order, miscibility, and electronic structure of $\text{Ag}\left(\text{Bi},\text{Sb}\right){\text{Te}}_2$ alloys and (Ag,Bi,Sb)Te precipitates in rocksalt matrix: A first-principles study

Using first-principles density-functional theory calculations and cluster expansion, we predict that ${\text{AgBiTe}}_2{\text{-AgSbTe}}_2$ alloys exhibit D4 cation order at all temperatures below melting and are fully miscible down to the room temperature and below. We also discuss the miscibility and ordering on the cation sublattice in quasiternary (Ag,Bi,Sb)Te alloys with general composition, within the subclass of structures with rocksalt topology (relevant for the case of coherent precipitates in a rocksalt matrix, e.g., in PbTe). The band structures of the ${\text{AgBiTe}}_2$ and ${\text{AgSbTe}}_2$ compounds and the evolution of the Fermi-surface topology at low hole dopings are presented. We use these results to refine the interpretation of the recent experimental measurements on naturally doped ${\text{AgSbTe}}_2$ samples reported by Jovovic and Heremans [Phys. Rev. B 77, 245204 (2008)] and present a simplified model of the band dispersion near the valence-band maximum.

Figure 1

(Color online) Gibbs triangle showing the $T=0$ stable structures along the ${\text{Ag}}_{1-\text{x}-\text{y}}{\text{Bi}}_{\text{x}}{\text{Sb}}_{\text{y}}\text{Te}$ isoplethal section. Panel (a) shows the thermodynamically stable (nonrocksalt) structures and panel (b) shows the structures stable under a constraint of rocksalt topology. The color-coded background in the online version of panel (b) shows the $T=0$ energetic cost (in meV/cation) of forming a rocksalt system compared to the nonrocksalt structures of panel (a).

Figure 2

(Color online) Calculated solid-state phase diagram of ${\text{AgBi}}_{1-\text{x}}{\text{Sb}}_{\text{x}}{\text{Te}}_2$ solid solutions, showing the order-disorder transition temperature $T_{\text{ord}}$ of the D4 ordered phase and a schematic upper bound on the $T_{\text{misc}}$ of the ${\text{AgBiTe}}_2{\text{-AgSbTe}}_2$ miscibility gap. Also marked are the experimental melting temperatures $T_{\text{m}}$ of pure bulk ${\text{AgSbTe}}_2$, ${\text{AgBiTe}}_2$, as well as that of PbTe, which could serve as a coherent matrix for $\text{Ag}\left(\text{Bi},\text{Sb}\right){\text{Te}}_2$ precipitates.

Figure 3

(Color online) Band structure of ordered ${\text{AgSbTe}}_2$ and ${\text{AgBiTe}}_2$. The radii of the color circles in the online version are proportional to the contributions from $\text{Te}5p$ (black), $\text{Ag}4d$ (blue), $\text{Ag}5s$ (green), and $\text{Bi}6p$ or $\text{Sb}5p$ (red). The energies are relative to the calculated value of the Fermi level.

Figure 4

(Color online) Dispersion of ${\text{AgSbTe}}_2$ valence bands, (a) for select directions and (b) within W-X-U facet of the Brillouin zone (shown in the inset). The dashed line in (a) corresponds to the measured Fermi level ${\epsilon }_F^{parab}$ (Ref. 20). The arrows highlight the band crossing points with conical energy dispersion. Band energies are relative to the GGA value of Fermi level ${\epsilon }_F^{\text{GGA}}$.

Figure 5

(Color online) The (100) cross sections of the Fermi pockets for different positions of ${\epsilon }_F$ relative to the VBM: (a) ${\epsilon }_{\text{VBM}}-{\epsilon }_F=15\text{meV}$, (b) 20 meV, (c) 60 meV, and (d) 160meV. The full X-centered facet of the Brillouin-zone boundary is shown as labeled in (a). In (c) and (d), the larger (light pink) Fermi pocket also fills the space occupied by the smaller (dark blue) pocket.

Figure 6

The extremal values of the Fermi cross-section area $A_F$ vs the presumed position of the Fermi level ${\epsilon }_F$. The bold symbols indicate the features giving the largest contribution to the de Haas-van Alphen signal (see Appendix). The wide horizontal gray lines correspond to the de Haas-van Alphen measurements of Ref. 20.

Figure 7

(Color online) Evolution of equilibrium compositions of rocksalt-constrained (Ag,Bi,Sb)Te alloys from 800 to 400 K for a finite set of chemical-potential combinations $\left\{{\mu }_{\text{Sb}},{\mu }_{\text{Bi}},{\mu }_{\text{Ag}}\right\}$. Each point indicates a composition achieved at some $\left\{{\mu }_{\text{Sb}},{\mu }_{\text{Bi}},{\mu }_{\text{Ag}}\right\}$; the regions between the points correspond either to intermediate chemical-potential values or to immiscible compositions.

Figure 8

The area $A_F$ of the Fermi-surface $⟨111⟩$ cross section vs the position $\delta x$ of the cross-sectioning plane, relative to the X point. The insets show the shape of the cross section(s) for a few values of $\delta x$. In cases of multiple cross sections seen in the insets, the main figure shows the net area of all cross sections in the vicinity of a given X point.