  Motivated by recent scanning tunneling microscopy/spectroscopy experiments on
probing single vacancies in black phosphorus, we present a theory for Fano
antiresonances induced by coupling between vacancy states and edge states of
zigzag phosphorene nanoribbons (zPNRs). To this end, in the first step, using
the tight-binding Hamiltonian, we obtain an analytic solution on the lattice
for the state associated to a single vacancy located in the bulk phosphorene
which shows a highly anisotropic localization in real space. For a finite
zigzag ribbon, in the absence of particle-hole symmetry, the localized state
induced by vacancies can couple to the wave functions of the edge states which
results in the formation of a new bound state. The energy of vacancy bound
state lies inside the quasi-flat band composed of edge states when the vacancy
locates sufficiently far away from the edge. Then, we employ the T-matrix
Lippmann-Schwinger approach to obtain an explicit analytical expression for the
scattering amplitude of the edge electrons of a zPNR by the presence of a
single vacancy which shows a Fano resonance profile with a tunable dip. We
demonstrate that varying the position of the vacancy produces substantially
different effects on the resonance width, resonance energy position, and the
asymmetry parameter of Fano line shape. Furthermore, the validity of the
theoretical descriptions is verified numerically by using the Landauer
approach.
