Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Figure 8 A simplistic model for describing the electronic structure in nanocrystals.Each valence band contributes a ladder of particle-in-a-sphere states for the hole.The optical transitions then occur between these hole states and the electron levelsarising from the conduction band. This model fails to predict the observed structuredue to mixing of the different hole ladders, as discussed in the text.quantum-dot hole state is a mixture of the three valence subbands (valencebandmixing) as well as particle-in-a-sphere envelope functions with angularmomentum L h and L h +2 (S-D mixing). The three independent ladders of holestates, as shown in Fig. 8, are coupled. The electron levels, which originate inthe simple conduction band that is largely unaffected by the valence-bandcomplexities, can be assumed to be well described by the particle-in-a-sphereladder. However, we will revisit this assumption below.When theory includes these effects, the size dependence observed in Fig.7 can be described. Using the approach of Efros et al. [55,77], in which theenergies of the hole states are determined by solving the Luttinger Hamiltonianand the electron levels are calculated within the Kane model, strongagreement with the data is obtained, as shown in Figs. 10 and 11. Figure 10compares theory with the lowest three transitions which exhibit simple sizedependentbehavior (i.e., no avoided crossings). Figure 11 shows the avoidedcrossing regions. The transitions can be assigned and labeled by modifiedparticle-in-a-sphere symbols which account for the valence-band mixingdiscussed earlier [9].Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.