Application of numerical exciton-wave-function calculations to the question of band alignment in ${\mathrm{S}\mathrm{i}/\mathrm{S}\mathrm{i}}_{1-x}{\mathrm{Ge}}_x$ quantum wells

We numerically calculate the two-particle ground-state wave functions for excitons in ${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ quantum wells in an effective mass model and demonstrate how oscillator strength, binding energy, and electron distribution vary with conduction-band offset and well width. Recombination energies for the two types of excitons involving electrons in different conduction-band valleys are compared. We point out that only due to the very different electron masses in growth direction for ${\Delta }_4$ and ${\Delta }_2$ valleys is it possible that the ${\Delta }_2$-heavy-hole exciton forms the ground state, as was reported in recent photoluminescence experiments at extremely low excitation powers [M. L. W. Thewalt et al., Phys. Rev. Lett. 79, 269 (1997)]. It is shown that those experiments can only be explained with a type-II offset for the ${\Delta }_4$ conduction band of about 40 meV, in contrast to the common assumption that those data would have proven an offset of at most 10 meV for $x=\mathrm{0.3.}\mathrm{}$