Designing atomic-scale resistive circuits in topological insulators through vacancy-induced localized modes

We demonstrate that vacancies can induce topologically protected localized electronic excitations within the bulk of a topological insulator, and when sufficiently close, give rise to one-dimensional propagating chiral bulk modes. We show that the dynamics of these modes can be effectively described by a tight-binding Hamiltonian, with the hopping parameter determined by the overlap of electronic wave functions between adjacent vacancies, accurately predicting the low-energy spectrum. Building on this phenomenon, we propose that vacancies in topological materials can be utilized to design atomic-scale resistive circuits, and estimate the associated resistance as a function of the vacancy distribution's geometric properties.