Conduction subbands in a GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As quantum well: Comparing different k⋅p models

The energy dispersion of the conduction subbands in a GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As superlattice is calculated by using a k⋅p Hamiltonian that includes different number of bands. The most accurate model includes the ${\mathrm{\Gamma }}_6^{\mathit{c}}$, ${\mathrm{\Gamma }}_7^{\mathit{v}}$, ${\mathrm{\Gamma }}_8^{\mathit{v}}$, ${\mathrm{\Gamma }}_7^{\mathit{c}}$, and ${\mathrm{\Gamma }}_8^{\mathit{c}}$ bands. The resulting subband dispersion is compared with that obtained when the coupling of the ${\mathrm{\Gamma }}_6^{\mathit{c}}$ band with the ${\mathrm{\Gamma }}_7^{\mathit{v}}$ and ${\mathrm{\Gamma }}_8^{\mathit{v}}$ and/or ${\mathrm{\Gamma }}_7^{\mathit{c}}$ and ${\mathrm{\Gamma }}_8^{\mathit{c}}$ bands is neglected. We also consider the 2×2 k⋅p Hamiltonian with terms up to the order ${\mathit{k}}^4$. The subband dispersions are analyzed quantitatively by fitting the numerical result to the analytical expression obtained with the invariant expansion technique. The subbands are found to be significantly different with the various models. The differences are ascribed to the different bulk band dispersions obtained with different k⋅p Hamiltonians and to the variation of the band parameters in the well and barrier materials.