Based on the minimum two-orbital model and the phase diagram recently proposed by Tai et al. (Europhys. Lett. textbf{103}, 67001(2013)) for both electron- and hole-doped 122 iron-based superconducting compounds, we use the Bogoliubov-de Gennes equations to perform a comprehensive investigation of the evolution of the Fermi surface (FS) topology in the presence of the collinear spin-density-wave (SDW) order as the doping is changed. In the parent compound, the ground state is the SDW order, where the FS is not completely gapped, and two types of Dirac cones, one electron-doped and the other hole-doped emerge in the magnetic Brillouin zone. Our findings are qualitatively consistent with recent angle-resolved photoemission spectroscopy and magneto-resistivity measurements. We also examine the FS evolution of both electron- and hole-doped cases and compare them with measurements, as well as with those obtained by other model Hamiltonians.
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