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

see that for all nanocrystal shapes, the excitation probability of the lowerjFj = 1 (F1 L ) exciton state, 2P1 L , decreases with size and that the upperjFj = 1 (F1 U ) gains its oscillator strength.This behavior can be understood by examining the spherically symmetriclimit. In spherical NCs, the exchange interaction leads to the formation oftwo exciton states—with total angular momenta 2 and 1. The ground state isthe optically passive state with total angular momentum 2. This state isfivefold degenerate with respect to the total angular momentum projection.For small nanocrystals, the splitting of the exciton levels due to the nanocrystalasymmetry can be considered as a perturbation to the exchange interaction(the latter scales as 1/a 3 ). In this situation, the wave functions of theF1 L , 0 L , and F2 exciton states turn into the wave functions of the opticallypassive exciton with total angular momentum 2. The wave functions of theF1 U and 0 U exciton states become those of the optically active exciton stateswith total angular momentum 1. Therefore, these three states carry nearly allof the oscillator strength.In large NCs, for all possible shapes, we can neglect the exchangeinteraction (which decreases as 1/a 3 ), and thus there are only two fourfolddegenerate exciton states (see Fig. 3). The splitting here is determined by theshape asymmetry and the intrinsic crystal field. In a system of randomlyoriented crystals, the excitation probability of both of these states is the same:P0 U þ 2PU 1 ¼ 2PL 1 ¼ 2KP2 =3 [12].In Fig. 5, we show these dependences for variously shaped CdTe NCswith a cubic lattice structure. It is necessary to note here that despite the factthat the exchange interaction drastically changes the structure and theoscillator strengths of the band-edge exciton, the linear polarization propertiesof the nanocrystal (e.g., the linear polarization memory effect) aredetermined by the internal and crystal shape asymmetries. All linear polarizationeffects are proportional to the net splitting parameter D and becomeinsignificant when D = 0.Our calculations show that the ground exciton state is always theoptically passive dark exciton independent of the intrinsic lattice symmetryand the shape of the NCs. In spherical NCs with the cubic lattice structure, theground exciton states has total angular momentum 2. It cannot be excited bythe photon and cannot emit the photon directly in the electric-dipoleapproximation. This limitation holds also for the hexagonal CdSe NCs. Theycannot emit or absorb photons directly, because the ground exciton state hasthe F2 angular momentum projections along the hexagonal axis. In small-sizeelongated NCs, the ground exciton state has a zero angular momentumprojection; however, it was also shown to be the optically forbidden darkexcitonstate.The radiative recombination of the dark exciton can only occur throughsome assisting processes that flip the electron-spin projection or change theCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.