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

In crystals for which the function u(b) is negative (this is, e.g., the case forCdSe for which b = 0.28 [15]), the net splitting decreases with size in prolate(l > 0) NCs. Even the order of the hole levels can change, with the jMj = 1/2state becoming the hole ground level for sufficiently small crystals [16]. Thiscan be qualitatively understood within a model of uncoupled A and B valencesubbands. In prolate crystals, the energy of the lowest hole quantum-size levelis determined by its motion in the plane perpendicular to the hexagonal axis.In this plane, the hole effective mass in the lowest subband A is smaller thanthat in the higher B subband [12]. Decreasing the size of the crystal causes ashift of the quantum-size level inversely proportional to both the effectivemass and the square of the NC radius. The shift is therefore larger for the Asubband than for the B subband and, as a result, it can change the order of thelevels in small NCs. In oblate (l < 0) crystals, where the levels are determinedby motion along the hexagonal axis, the B subband has the smaller mass.Hence, the net splitting increases with decreasing size and the states maintaintheir original order.The eight-fold degeneracy of the spherical band-edge exciton is alsobroken by the electron–hole-exchange interaction which mixes different electronand hole spin states. This interaction can be described by the followingexpression [13,17]:ˆ H exch ¼ 2 3 e exchða 0 Þ 3 dðr e r h ÞsJ ð11Þwhere s is the electron Pauli spin-1/2 matrix, J is the hole spin-3/2 matrix, a 0 isthe lattice constant, and e exch is the exchange strength constant. In bulkcrystals with cubic lattice structure, this term splits the eightfold degenerateground exciton state into a fivefold degenerate optically passive state withtotal angular momentum 2 and a threefold degenerate optically active statewith total angular momentum 1. This splitting can be expressed in terms of thebulk exciton Bohr radius, a ex :tx ST ¼8 33 pa0e exch ð12Þa exIn bulk crystals with hexagonal lattice structure, this term splits the excitonfourfold degenerate ground state into a triplet and a single state, separated bytx ST ¼2 3 a0e exch ð13Þpa exEquations (12) and (13) allow one to evalute the exchange strength constant.In CdSe crystals, where tx ST = 0.13 meV [18], a value of e exch = 450 meV isobtained using a ex = 56 A˚ .Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.