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

So far, this treatment has completely ignored the Coulombic attractionbetween the electron and the hole, which leads to excitons in the bulk material.Of course, the Coulombic attraction still exists in the nanocrystal. However,how it is included depends on the confinement regime [2]. In the strongconfinement regime, another approximation, the strong confinement approximation,is used to treat this term. According to Eq. (5), the confinementenergy of each carrier scales as 1/a 2 . The Coulomb interaction scales as 1/a. Insufficiently small crystallites, the quadratic confinement term dominates.Thus, in the strong confinement regime, the electron and hole can be treatedindependently and each is described as a particle in a sphere. The Coulombterm may then be added as a first-order energy correction, E c . Therefore,using Eqs (3), (5), and (9), the electron–hole pair (ehp) states in nanocrystalsare written asC ehp ð ! r e ; ! r h Þ ¼ C e ð ! rÞC h ð ! r h Þ¼ u c f e ð ! r e Þu v f h ð ! r h Þj Le ðk ne ;L er e ÞY m eL¼ C u ej Lh ðk nh ;L hr h ÞY m hLc u hvr er hð11Þwith energiesE ehp ðn h L h n e L e Þ ¼ E g þ t22a 2( )B 2 n h ;L hm v effþ B2 n e ;L em c effE cð12ÞThe states are labeled by the quantum numbers n h L h n e L e . For example, thelowest-pair state is written as 1S h 1S e . For pair states with the electron in the1S e level, the first-order Coulomb correction, E c , is 1.8e 2 /qa, where q isthe dielectric constant of the semiconductor [4]. Equations (11) and (12) areusually referred to as the particle-in-a-sphere solutions to the nanocrystalspectrum.C. Optical Transition ProbabilitiesThe probability to make an optical transition from the ground state, j0i, to aparticular electron–hole pair state is given by the dipole matrix elementP ¼ hCehp ! e pˆ0i 2ð13Þwhere ! e is the polarization vector of the light and ˆp is the momentumoperator. In the strong confinement regime, where the carriers are treatedCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.