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

Figure 13 Energy level diagram describing the exciton fine structure. In thespherical model, the band-edge exciton (1S 3/2 1S e ) is eightfold degenerate. Thisdegeneracy is split by the nonspherical shape of the dots, their hexagonal (wurtzite)lattice, and the exchange interaction.of Fig. 13, the correlation diagram between these two limits is shown. Whenboth effects are included, the good quantum number is the projection of Nalong the unique crystal axis, N m . The five sublevels are then labeled by jN m j:one sublevel with jN m j = 2, two with jN m j = 1, and two with jN m j = 0. Levelswith jN m j > 0 are twofold degenerate.To include these effects into the theory, the anisotropy and exchangeterms can be added as perturbations to the spherical model [10]. Figure 14shows the calculated size dependence of the exciton fine structure. The fivesublevels are labeled by jN m j with superscripts to distinguish upper (U) andlower (L) sublevels with the same jN m j. Their energy, relative to the 1 Lsublevel, is plotted versus effective radius, which is defined asa eff ¼ 1 2 ðb2 cÞ 1=3ð24Þwhere b and c are the short and long axes of the nanocrystal, respectively. Theenhancement of the exchange interaction with decreasing nanocrystal size isclearly evident in Fig. 14. Conversely, with increasing nanocrystal size, thesublevels converge upon the bulk A–B splitting, as expected.E. The Dark ExcitonAt first glance, one may feel that the exciton fine structure is only a smallrefinement to the theoretical model with no real impact on the properties ofCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.