Optical transitions and the nature of Stokes shift in spherical CdS quantum dots

Resonant Stokes shift observed in CdS quantum dots (QDs) has been previously studied theoretically using {\bf k$\cdot$p} approach. The large values of measured Stokes shift along with the structure of the excitonic levels obtained by the {\bf k$\cdot$p} calculations have suggested an optically forbidden $P$ envelope valence state, thus forming a spatial symmetry induced ``dark exciton'' in CdS QDs, in contrast with the spin-forbidden exchange interaction induced ``dark exciton'' found in CdSe QDs. Since the {\bf k$\cdot$p} method has been known to incorrectly predict the energy levels in other QDs, here we apply {\sl ab initio} accuracy methods to study this problem. Using the LDA-based charge patching method to generate the Hamiltonian, combined with the folded spectrum method to solve the single particle states of thousand-atom nanostructures, we find that the top of the valence band state is $S$-like, thus optically bright, in contrast with all the previous {\bf k$\cdot$p} calculations. Our results also indicate the range of applicability of the {\bf k$\cdot$p} method. The calculated electron-hole exchange splitting suggests that the spin-forbidden valence state may explain the nature of the ``dark exciton'' in CdS quantum dots.