Formalism for tunneling of mixed-symmetry electronic states: Application to electron and hole tunneling in direct- and indirect-band-gap GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As structures

A formalism is presented to determine the tunneling properties of semiconductor heterostructures by studying the time evolution of multiband electronic states of mixed central-cell symmetry. The time evolution of the multiband wave function is determined by numerically solving the Schrödinger equation with use of a unitary approximation of the time-evolution operator valid for infinitesimal time steps. The valence-band states are studied with use of a four-band k⋅p approach, and results presented for resonant tunneling of holes in coupled quantum wells. The tunneling of an electron wave packet from a GaAs well through direct- and indirect-band-gap ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As barriers is studied with use of the tight-binding representation for the conduction-band states in an eight-element (${\mathit{sp}}^3$) basis. The strong suppression of tunneling for the indirect-band-gap case is explained by the central-cell-symmetry variation in real space.