Operator ordering and interface-band mixing in the Kane-like Hamiltonian of lattice-matched semiconductor superlattices with abrupt interfaces

Abstract

A Kane-like envelope function Hamiltonian is derived for the ${\ensuremath{\Gamma}}_{15}$ valence and ${\ensuremath{\Gamma}}_{1}$ conduction states of lattice-matched, semiconductor superlattice structures, with metallurgically abrupt interfaces. The local microscopic potential is treated as a weak perturbation on that of a reference crystal and is expressed in terms of a one-dimensional profile function, $G(z)$, which modulates the difference between the potentials of the well and barrier materials. In contrast to many previous treatments, all terms up to order $\ensuremath{\ell}=2$ in $\ensuremath{\delta}\overline{V}\ensuremath{\cdot}{(\overline{k}a)}^{\ensuremath{\ell}}$ are included, where $\ensuremath{\delta}\overline{V}$ is the typical band offset, $\ensuremath{\hbar}\overline{k}$ is the average momentum modulus of the envelope function, and $a$ is the bulk lattice parameter. Far from the interfaces, the Hamiltonian is identical to the familiar bulk Kane Hamiltonian, with the standard bulk parameters. However, the operator ordering in the valence band is revised from the commonly used Burt scheme. An operator ordering scheme has also been derived for the linear-$k$ $P$ terms that couple conduction and valence states. Expressions have been derived for the $\ensuremath{\delta}$ functionlike, and derivative of a $\ensuremath{\delta}$ functionlike, interface terms. These are off-diagonal and diagonal, respectively, in common atom superlattices like $\text{GaAs}/{\text{Al}}_{x}{\text{Ga}}_{1\ensuremath{-}x}\text{As}$, where the antisymmetric contribution to ${G}^{\ensuremath{'}}(z)$ is expected to be small. For superlattices with no common atom, additional interface terms are introduced. If the difference in the spin-orbit splitting energy for the two superlattice materials is comparable with the valence-band offset, then relativistic corrections can introduce many more, weak interface contributions. Part of the relativistic interface matrix has been derived, which includes the most significant terms. Finally, a scheme is proposed for reducing the number of independent Luttinger parameters required, when using the Hamiltonian to fit experimental spectral data.