Analysis of the crystal structure of alloys of the [(Ge,Sn,Pb)(Te,Se)]$_{m}$[(Bi,Sb)$_{2}$(Te,Se)$_{3}]_{n}$ ($m,n$ = 0, 1, 2, $\dots$) family within the theory of closely packed spheres

The processes of the formation of complex crystal structures–layered crystals of [(Ge,Sn,Pb)(Te,Se)]$_{m}$[(Bi,Sb)$_{2}$(Te,Se)$_{3}]_{n}$ ($m,n$ = 0, 1, 2, $\dots$) ternary alloys with different symmetries (space groups $R\bar3m$, $P\bar3m_{1}$, and $P2_{1}/m$)–are investigated within the theory of closely packed spheres (CPS theory). The structures of layered binary alloys of the Bi$_2$Te$_3$ $(R\bar3m)$ and PbTe ($Fm\bar3m$ and $R3m$) types are found to be related and formed based on the $s$3 triplet with cubic symmetry $(Fm\bar3m)$, which is constructed by chalcogen (Se or Te) atoms.