Derivation of 12- and 14-bandk·pHamiltonians for dilute bismide and bismide-nitride semiconductors

Using an $sp³s^*$ tight-binding model we demonstrate how the observed strong bowing of the band gap and spin-orbit-splitting with increasing Bi composition in the dilute bismide alloy GaBi$_x$As$_1-x$ can be described in terms of a band-anticrossing interaction between the extended states of the GaAs valence band edge and highly localised Bi-related resonant states lying below the GaAs valence band edge. We derive a 12-band $\textbf{k}\cdot\textbf{p}$ Hamiltonian to describe the band structure of GaBi$_x$As$_1-x$ and show that this model is in excellent agreement with full tight-binding calculations of the band structure in the vicinity of the band edges, as well as with experimental measurements of the band gap and spin-orbit-splitting across a large composition range. Based on a tight-binding model of GaBi$_x$N$_y$As$_1-x-y$ we show that to a good approximation N and Bi act independently of one another in disordered GaBi$_x$N$_y$As$_1-x-y$ alloys, indicating that a simple description of the band structure is possible. We present a 14-band $\textbf{k}\cdot\textbf{p}$ Hamiltonian for ordered GaBi$_x$N$_y$As$_1-x-y$ crystals which reproduces accurately the essential features of full tight-binding calculations of the band structure in the vicinity of the band edges. The $\textbf{k}\cdot\textbf{p}$ models we present here are therefore ideally suited to the simulation of the optoelectronic properties of these novel III-V semiconductor alloys.