Exploring the Electronic Structures Across a Series of Topological Superstructures

Three-dimensional topological insulators, which exhibit bulk insulating behavior along with symmetry-protected metallic surface states, are classified by four Z₂ invariants ν_i​​​​ (i=0,1,2,3). Strong topological materials, with a nontrivial strong invariant (ν₀=1) and trivial weak invariants (ν_i​​​​=1 for i=1,2,3), have surface states that are robust against perturbations on all surfaces whereas weak topological materials, with a trivial strong invariant but at least one nontrivial weak invariant, exhibit surface states on select surfaces and host 1D helical edge modes along line defects. Most topological materials fall into one of the two classes, but it is possible for a material to have nontrivial strong and weak invariants, combining robust surface states with helical 1D defect channels. A homologous series of topological superlattices is predicted to have ν₀=1 and weak topological indices that vary between trivial and nontrivial depending on the superstructure composition, providing a unique platform to understand and engineer topological states. Using angle-resolved photoemission spectroscopy, we explore the evolution of the bulk and surface electronic structure across the homologous series, offering insights into superstructure engineering of topological materials.