The concept of topological Fano resonance, characterized by an ultrasharp asymmetric line shape, is a promising candidate for robust sensing applications due to its sensitivity to external parameters and immunity to structural disorder. In this study, the vacancy-induced topological Fano resonance in a nanoribbon made up of a hexagonal lattice with armchair sides is examined by introducing several on-site vacancies, which are deliberately created at regular distances, along a zigzag chain that stretches across the width of the ribbon. The presence of the on-site vacancies can create localized energy states within the electronic band structure, leading to
the formation of an impurity band, which can result in Fano resonance phenomena by forming a conductivity channel between the edge modes on both armchair sides of the ribbon. Consequently, an ultracompact tunable on-chip integrated topological
Fano resonance derived from the graphene-based nanomechanical phononic crystals is proposed. The Fano resonance arises from the interference between topologically protected even and odd edge modes at the interface between trivial and nontrivial
insulators in a nanoribbon structure governed by the Kane-Mele model describing the quantum spin Hall e®ect in hexagonal lattices. The simulation of the topological Fano resonance is performed analytically using the Lippmann-Schwinger scattering
formulation. One of the advantages of the present study is that the related calculations are carried out analytically, and in addition to the simplicity and directness, it reproduces the results obtained from the Landauer-BÄuttiker formulation very well both
quantitatively and qualitatively. The ¯ndings open up possibilities for the design of highly sensitive and accurate robust sensors for detecting extremely tiny forces, masses, and spatial positions.
