Simplified four-band k · p model to consider k-dependent band-mixing effects in electron intersubband scattering: Application to quantum wells and quantum cascade lasers

Recently, a first-order k⋅p Kane model was introduced to include k-dependent band-mixing in intersubband scattering [Mac et al., Phys. Rev. B 110, 165304 (2024)]. This work proposes to reduce the eight-component eigenstates in the scattering matrix element to four-component eigenstates to improve computational efficiency. This is achieved by removing Rashba spin–orbit (SO) coupling terms from the k-dependent eigenfunctions via a change-of-base and subsequent high-bandgap approximation. Unlike the eight-band model, the four-band model can calculate the total scattering rate without needing to treat spin-conserving and spin-flip transitions separately. Furthermore, compact analytical solutions exist for the scattering angle integration in longitudinal-optical (LO) phonon scattering. Band-mixing effects due to Rashba SO coupling are shown to have a minimal effect even in biased asymmetric quantum well structures. This includes nine mid-infrared quantum cascade lasers (QCLs) ranging from λ∼3.3–15.7 μm. In all these structures, scattering rates differ by ≤1% between the two models. Furthermore, the contrast of the scattering rate with respect to the spin of the initial eigenstate remains minimal. Thus, for scalar scattering, the compact four-band model can be used in place of the eight-band model to treat band-mixing in scattering without a significant loss in accuracy. Finally, we reiterate the importance of incorporating k⋅p-modified scattering in QCL design. The k-dependent wavefunction confinement effect, which is seldom mentioned in the literature, is shown to significantly impact the spatial overlap between two states and therefore their associated scattering.