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Using an $sp^{3}s^{*}$ tight-binding model we demonstrate how the observedstrong bowing of the band gap and spin-orbit-splitting with increasing Bicomposition in the dilute bismide alloy GaBi$_{x}$As$_{1-x}$ can be describedin terms of a band-anticrossing interaction between the extended states of theGaAs valence band edge and highly localised Bi-related resonant states lyingbelow the GaAs valence band edge. We derive a 12-band$\textbf{k}\cdot\textbf{p}$ Hamiltonian to describe the band structure ofGaBi$_{x}$As$_{1-x}$ and show that this model is in excellent agreement withfull tight-binding calculations of the band structure in the vicinity of theband edges, as well as with experimental measurements of the band gap andspin-orbit-splitting across a large composition range. Based on a tight-bindingmodel of GaBi$_{x}$N$_{y}$As$_{1-x-y}$ we show that to a good approximation Nand Bi act independently of one another in disorderedGaBi$_{x}$N$_{y}$As$_{1-x-y}$ alloys, indicating that a simple description ofthe band structure is possible. We present a 14-band$\textbf{k}\cdot\textbf{p}$ Hamiltonian for orderedGaBi$_{x}$N$_{y}$As$_{1-x-y}$ crystals which reproduces accurately theessential features of full tight-binding calculations of the band structure inthe vicinity of the band edges. The $\textbf{k}\cdot\textbf{p}$ models wepresent here are therefore ideally suited to the simulation of theoptoelectronic properties of these novel III-V semiconductor alloys.

Derivation of 12- and 14-band $\textbf{k}\cdot\textbf{p}$ Hamiltonians for dilute bismide and bismide-nitride semiconductors