Linearized band gap evolution in topologically gapped alloys: a case study of KZnSb_1-xBi_x

Band gap is the most fundamental property underlying the device performance of a gapped crystalline material, either a semiconductor/insulator or a topological insulator. A well-established bowing curve characterizes the concentration dependence of trivial band gap in semiconductor alloys, while the concentration dependence of nontrivial gap in topological alloys remains largely unknown. This question is especially important because the topological alloys have usually a narrow gap which can be easily tuned by chemical composition where not only the size but also the “sign” of gap can be changed due to band inversion. We have systematically investigated the band gap (E_g) evolution of KZnSb_1-xBi_x alloy as a function of alloy concentration (x), using first-principles calculations within virtual crystal approximation. Interestingly, we found that the gap evolves linearly within each topological phase region (normal insulator, Dirac semimetal,  and topological crystalline insulator) separated by singular transition points, i.e., the band crossing and reopening points. This ideal linear topological gap function, in analogy to mixing enthalpy of ideal solution, is revealed to be driven by the interplay of chemical potential (on-site energy) and spin-orbit coupling of mixing, which differs from the parabolic bowing gap function of semiconductors, in analogy to regular solution, driven only by chemical potential of mixing.  We further show that the DSM phase in the range of x=0.23-0.45 displays non-trivial 1^st- and 2^nd-order band topology, with non-trivial surface states and higher order Fermi arc.