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

the six-band Luttinger Hamiltonian. Rather, the experiment suggests that anadditional nonparabolicity is present in the bands that is not accounted foreven by the eight-band model. However, the question remains: What thecause of this nonparabolicity?The observation that the theory correctly predicts the transition energieswhen plotted relative to the first excited state is an important clue.Because most of the low-lying optical transitions share the same electron level(1S e ), Fig. 20 implies that the theory is struggling to predict the size dependenceof the strongly confined electrons. By plotting relative to the energy ofthe first excited state, Figs. 10 and 11 remove this troubling portion [8]. Then,the theory can accurately predict the transitions relative to this energy.Because the underlying cause cannot be the mixing between theconduction and valence band, we must look for other explanations. Althoughthe exact origin is still unknown, it is easy to speculate about several leadingcandidates. First, a general problem exists in how to theoretically treat thenanocrystal interface. In the simple particle-in-a-sphere model [Eq. (2)], thepotential barrier at the surface was treated as infinitely high. This is theoreticallydesirable because it implies that the carrier wave function goes to zero atthe interface. Of course, in reality, the barrier is finite and some penetration ofthe electron and hole into the surrounding medium must occur. This effectshould be more dramatic for the electron, which is more strongly confined. Topartially account for this effect, the models used to treat CdSe and InAsnanocrystals incorporated a finite ‘‘square well’’ potential barrier, V e , for theelectron. (The hole barrier was still assumed to be infinite.) However, inpractice, V e became simply a fitting parameter to better correct for deviationsin Fig. 20. In addition, the use of a square potential barrier is not a rigoroustreatment of the interface. In fact, how one should analytically approach suchan interface is still an open theoretical problem. The resolution of this issue forthe nanocrystal may require more sophisticated general boundary conditiontheories that have recently been developed [95].A second candidate to explain Fig. 20 is the simplistic treatment of theCoulomb interaction, which is included only as a first-order perturbation.This approach not only misses additional couplings between levels but also, asrecently pointed out by Efros et al. [41], ignores the expected size dependencein the dielectric constant. The effective dielectric constant of the nanocrystalshould decrease with decreasing size. This implies that the perturbativeapproach underestimates the Coulomb interaction. Unfortunately, this effecthas not yet been treated theoretically.Finally, one could also worry, in general, about the breakdown of theeffective mass and the envelope function approximations in extremely smallnanocrystals. As discussed in Section II.B, we require the nanocrystals to bemuch larger than the lattice constant of the semiconductor. In extremely smallCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.