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Transition between the topologically trivial and non-trivial phase ofPb_{1-x}Sn_{x}Te alloy is driven by the increasing content $x$ of Sn, or by thehydrostatic pressure for $x<0.3$. We show that a sharp border between these twotopologies exists in the Virtual Crystal Approximation only. In more realisticmodels, the Special Quasirandom Structure method and the supercell method (withaveraging over various atomic configurations), the transitions are broadened.We find a surprisingly large interval of alloy composition, $0.3<x<0.6$, inwhich the energy gap is practically vanishing. A similar strong broadening isalso obtained for transitions driven by hydrostatic pressure. Analysis of theband structure shows that the alloy broadening originates in splittings of theenergy bands caused by the different chemical nature of Pb and Sn, and by thedecreased crystal symmetry due to spatial disorder. Based on our results of abinitio and tight binding calculations for Pb_{1-x}Sn_{x}Te we discuss differentcriteria of discrimination between trivial and nontrivial topology of the bandstructure of alloys.

Alloy broadening of the transition to the non-trivial topological phase of Pb_{1-x}Sn_{x}Te