Energy-dependent modulations in the local density of states in the under-doped and optimally doped NaFe$_{1-x}$Co$_{x}$As

Motivated by recent scanning tunneling microscopy experiment investigating the quasiparticle interference (QPI) patterns in NaFe$_{1-x}$Co$_{x}$As, we investigate the energy-dependent modulation of local density of states induced by a weak defect using the first-order T-matrix approximation. In the under-doped sample with spin-density-wave ordering, the electrons disperse along the antiferromagneitc direction but remain static along the ferromagnetic direction. The optimally doped sample exhibits square-like QPI patterns. The corresponding QPI in $\textbf{k}$ space is also presented. These numerical results exhibit essential features as those measured by the experiment.