Alloy broadening of the transition to the nontrivial topological phase of ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$

Transition between the topologically trivial and nontrivial phase of ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$ 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 ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$ we discuss different criteria of discrimination between trivial and nontrivial topology of the band structure of alloys.

Figure 1

Lattice constant dependence of (a) the PbTe band gap, (b) parameter $C_{\text{Te}}$, and topological indices (c) MCN, and (d) SCN.

Figure 2

Energy levels for a 200-atom PbTe layer oriented in the [001] direction for different lattice constant: (a) $a/a_0=0.960$, (b) $a/a_0=0.965$, (c) $a/a_0=0.970$, and (d) $a/a_0=0.975$.

Figure 3

Dependence of (a) the energy gap at the $L$ point and (b) the mirror Chern number on the Sn content $x$ for ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Te}$ in the VCA.

Figure 4

Band structures of ${\mathrm{Pb}}_4{\mathrm{Te}}_4$ (a) and of ${\mathrm{Pb}}_3{\mathrm{Sn}}_1{\mathrm{Te}}_4$ (b) for topologically trivial (upper row) and nontrivial (lower row) cases. Blue and red colors correspond to the valence and conduction states, respectively. The zero energy corresponds to the energy of highest valence band at the $L_{\mathrm{gap}}$ point (see text). In the left column, the bands are additionally degenerate due to the Brillouin zone folding, and the degeneracies are given in parentheses.

Figure 5

Energy levels of the eight highest valence bands and the eight lowest conduction bands at the $L_{\mathrm{gap}}$ point of the Brillouin zone relative to one of the cation-related band levels. The continuous and broken lines correspond to levels which wave functions are built mainly from $p$ orbitals of anions and cations, respectively. The numbers near the lines give the degeneracy of the levels ($2+4$ means that we have two lines, invisible in this scale, with the degeneracies 2 and 4, respectively). The thin vertical lines are the borders between topologically nontrivial, transition, and topologically trivial regions of the lattice constant. Notice that for ${\mathrm{Pb}}_4{\mathrm{Te}}_4$ the width of the transition region is zero.

Figure 6

Band structure characteristics of ${\mathrm{Pb}}_3{\mathrm{Sn}}_1{\mathrm{Te}}_4$. Dependence on the lattice constant of (a) minimal energy gap along $\mathrm{\Gamma }\to L_{\mathrm{gap}}$ direction and $E_{\mathrm{gap}}$ at the $L_{\mathrm{gap}}$, (b) contributions of pseudoatomic $p$ orbitals to the valence band wave functions at $L_{\mathrm{gap}}$, and (c) the mirror Chern number for valence bands.

Figure 7

Band energy levels at the $L_{\mathrm{gap}}$ point for anions and cations relative to one of the cation-related band levels (a) and the spin Chern number (b) as the functions of the lattice constant for one of the configurations for 16-atom supercell containing three Sn atoms. The thin vertical lines are the borders between topologically nontrivial, transition, and topologically trivial regions of the lattice constant.

Figure 8

(a) Minimal energy gaps on the [111] direction for 20 randomly chosen tin configurations (points) and the average (solid line). (b) Minimal energy gaps on the [111] direction for the best SQS tin configurations (solid line), for the configurations of minimal energies chosen from 10 best SQS (line with points), and for the VCA (dotted line).