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Recently the presence of topologically protected edge-states inBi$_{14}$Rh$_3$I$_9$ was confirmed by scanning tunnelling microscopyconsolidating this compound as a weak 3D topological insulator (TI). Here, wepresent a density-functional-theory-based study on a family of TIs derived fromthe Bi$_{14}$Rh$_3$I$_9$ parent structure via substitution of Ru, Pd, Os, Irand Pt for Rh. Comparative analysis of the band-structures throughout theentire series is done by means of a unified minimalistic tight-binding modelthat evinces strong similarity between the quantum-spin-Hall (QSH) layer inBi$_{14}$Rh$_3$I$_9$ and graphene in terms of $p_z$-molecular orbitals.Topologically non-trivial energy gaps are found for the Ir-, Rh-, Pt- andPd-based systems, whereas the Os- and Ru-systems remain trivial. Furthermore,the energy position of the metal $d$-band centre is identified as the parameterwhich governs the evolution of the topological character of the band structurethrough the whole family of TIs. The $d$-band position is shown to correlatewith the chemical bonding within the QSH layers, thus revealing how thechemical nature of the constituents affects the topological band character.

Correlation between topological band character and chemical bonding in a $\mathbf{Bi_{14}Rh_{3}I_{9}}$-based family of insulators