We employ first-principles electronic structure calculations to explore the effects of Bi doping at the As site of ${\mathrm{BaFe}}_2{\mathrm{As}}_2$, predicting significant changes in band topology and surface states. The negative chemical pressure induced by Bi doping, which has also been observed experimentally, is shown to drive distinct Lifshitz transitions that effectively act as electron doping, modifying the Fermi surface topology. These theoretical predictions suggest potential impacts on superconducting properties, with optimal superconductivity likely near $x=0.01$. Additionally, Dirac-cone-type surface states at the $\overline{\mathrm{\Gamma }}$ point progressively shift towards the Fermi level with increasing Bi concentration. This evolution is pivotal for the realization of topological superconducting surface states, the observation of Majorana zero-energy modes, and potential practical applications, including quantum computation. Using symmetry indicators derived from wavefunction symmetry at high-symmetry momenta, we confirm a nontrivial topological phase across all studied doping levels, from $x=0.005$ to $x=0.05$. The nontrivial nature of the ${\mathbb{Z}}_2$-topological invariant, while a topological phase transition corresponding to a higher topological index is predicted at $x=0.03$. These findings highlight that Bi-doped ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ retains a nontrivial topological phase throughout the studied doping range and suggest its potential as a platform for exploring Majorana zero-energy modes in its low-doping superconducting state.

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• Iron-based superconductors
• Pnictides
• Band structure methods
• Density functional theory
• Superconductors
• Symmetries in condensed matter