Unveiling topological phase complex through structurally tunning bismuth halides

The synergy of geometry, symmetry, and topology has profound implications especially in the study of topological materials. While strong topological insulators are not rare experimentally, the realizations of weak and higher-order topological insulators in solids are extremely challenging. Our recent studies have shown that quasi-one-dimensional Bi₄X₄ (X = Br, I) is an ideal system to overcome all such challenges, because of its weak interlayer coupling, large intrinsic gap, band simplicity near Fermi level, and multiple clean cleavage surfaces. Here we carry out a systematic study of this fertile candidate system through strategic doping of Br on I sites to tune the bulk crystal structural and achieve seven different structural phases. Strikingly, the intricate stacking manipulation leads to a cascade of topological phases including strong, weak, higher-order, and trivial insulators. This study suggests that quasi-one-dimensional Bi₄X₄ is a superior semiconducting system for exploring topologically nontrivial phases and their potential applications.