Alloy broadening of the transition to the nontrivial topological phase of Pb1−xSnxTe

Transition between the topologically trivial and nontrivial phase of ${\mathrm{Pb}}_{1\ensuremath{-}x}{\mathrm{Sn}}_x\mathrm{Te}$ alloy is driven by the increasing content $x$ of Sn, or by the hydrostatic pressure for $x<0.3$. We show that a sharp border between these two topologies exists in the virtual crystal approximation only. In more realistic models, the special quasirandom structure method and the supercell method (with averaging over various atomic configurations), the transitions are broadened. We find a surprisingly large interval of alloy composition, $0.3<x<0.5$, in which the energy gap is practically vanishing. A similar strong broadening is also obtained for transitions driven by hydrostatic pressure. Analysis of the band structure shows that the alloy broadening originates in splittings of the energy bands caused by the different chemical nature of Pb and Sn, and by the decreased crystal symmetry due to spatial disorder. Based on our results of ab initio and tight-binding calculations for ${\mathrm{Pb}}_{1\ensuremath{-}x}{\mathrm{Sn}}_x\mathrm{Te}$ we discuss different criteria of discrimination between trivial and nontrivial topology of the band structure of alloys.