Long- and short-range electron–hole exchange interaction in different types of quantum dots

The electron–hole (e–h) exchange interaction leads to the splitting of the exciton into a pair of bright and a pair of dark states. This bright–dark—or singlet–triplet—exciton splitting was historically calculated as the sum of a long-range (LR) and a short-range (SR) component. Using a numerical atomistic approach, we are able to calculate the exchange integrals as a function of the e–h range of interaction S, revealing the 'internal' structure of the integrals. We apply this procedure to thickness-fluctuation GaAs/AlGaAs quantum dots (QDs), self-assembled InAs/GaAs QDs and colloidal InAs QDs. We find a heterogeneous situation, where the SR component contributes ∼10, ∼20–30 and ∼20–50% to the total e–h exchange splitting, which is in the range of 10, 100 and 10 000 μeV, for the three types of QDs, respectively. The balance between SR and LR is found to depend critically on the size, shape and type of structure. For all types of QDs we find, surprisingly, a range of interaction, close to the physical dimension of the structures, contributing to a reduction of the integral's magnitude. These results highlight the complexity of the exchange interaction, warning against simplified models, and establish the basic features of the nature and origin of dark–bright excitonic splitting in QDs.