  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 effect 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-B\"{u}ttiker formulation very well both
quantitatively and qualitatively. The findings open up possibilities for the
design of highly sensitive and accurate robust sensors for detecting extremely
tiny forces, masses, and spatial positions.
