Dark excitons due to direct Coulomb interactions in silicon quantum dots

Electron-hole exchange interactions can lead to spin-forbidden “dark” excitons in direct-gap quantum dots. Here, we explore an alternative mechanism for creating optically forbidden excitons. In a large spherical quantum dot made of a diamond-structure semiconductor, the symmetry of the valence band maximum (VBM) is $t_2.$ The symmetry of the conduction band minimum (CBM) in direct-gap material is $a_1,$ but for indirect-gap systems the symmetry could be (depending on size) $a_1,$ e, or $t_2.$ In the latter cases, the resulting manifold of excitonic states contains several symmetries derived from the symmetries of the VBM and CBM (e.g., $t_2\times t_2{=A}_1{+E+T}_1{+T}_2$ or $t_2\times {e=T}_1{+T}_2).\mathrm{}$ Only the $T_2$ exciton is optically active or “bright,” while the others $A_1,$ E, and $T_1$ are “dark.” The question is which is lower in energy, the dark or bright. Using pseudopotential calculations of the single-particle states of Si quantum dots and a direct evaluation of the screened electron-hole Coulomb interaction, we find that, when the CBM symmetry is $t_2,$ the direct electron-hole Coulomb interaction lowers the energy of the dark excitons relative to the bright $T_2$ exciton. Thus, the lowest energy exciton is forbidden, even without an electron-hole exchange interaction. We find that our dark-bright excitonic splitting agrees well with experimental data of Calcott et al., Kovalev et al., and Brongersma et al. Our excitonic transition energies agree well with the recent experiment of Wolkin et al. In addition, and contradicting simplified models, we find that Coulomb correlations are more important for small dots than for intermediate sized ones. We describe the full excitonic spectrum of Si quantum dots by using a many-body expansion that includes both Coulomb and exchange electron hole terms. We present the predicted excitonic spectra.