An analytic approximation with a wide range of applicability for band-to-band radiative transition rates in direct, narrow-gap semiconductors

Good approximations for the radiative transition rates, conduction band to heavy-hole band and conduction band to light-hole band, which are continuous over a wide range of temperature and electron and hole Fermi levels (degenerate and non-degenerate) and in which photon multiplication may be included, are developed for direct gap III - V semiconductors. They are based on the band structure and wavefunctions of the simple isotropic three-band approximation to Kane's band structure and depend solely on the effective masses at the zone centre, the energy gap and the dielectric constant. They are compared with more accurate calculations based on the diagonalization of the matrix of the four-band model with higher and lower bands taken into account by perturbation theory. Application to (0.0 < x < 0.25) and (0.17 < x < 0.26) shows that over the whole range of materials, Fermi levels (-0.25 to 0.40 eV) and temperatures (80 - 400 K) the agreement is never worse than 20% and usually considerably better. Results are given in detail for InSb and . It is also shown that in degenerate material, when the penetration of the hole Fermi level into the valence band is greater than that of the electron Fermi level into the conduction band, the transition rate to the light-hole band can exceed that to the heavy-hole band. These approximations are eminently suitable for use in modelling semiconductor effects and devices.