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

electron and hole to be treated as if they were free particles, but with a differentmass.However, to utilize the effective mass approximation in the nanocrystalproblem, the crystallites must be treated as a bulk sample. In other words, weassume that the single-particle (electron or hole) wave function can be writtenin terms of Bloch functions [Eq. (6)] and that the concept of an effective massstill has meaning in a small quantum dot. If this is reasonable, we can utilizethe parabolic bands in Fig. 1a to determine the electron levels in the nanocrystal,as shown in Fig. 1b. This approximation, sometimes called theenvelope function approximation [24,25], is valid when the nanocrystal diameteris much larger than the lattice constant of the material. In this case, thesingle-particle (sp) wave function can be written as a linear combination ofBloch functionsC sp ð ! rÞ ¼ X kC nk u nk ð ! rÞexpð ! k ! rÞð8Þwhere C nk are expansion coefficients which ensure that the sum satisfies thespherical boundary condition of the nanocrystal. If we further assume that thefunctions u nk have a weak k dependence, then Eq. (8) can be rewritten asC sp ð ! rÞ ¼ u n0 ð ! rÞ X kC nk expði ! k ! rÞ ¼ u n0 ð ! rÞf sp ð ! rÞð9Þwhere f sp ð ! rÞ is the single-particle envelope function. Because the periodicfunctions u n0 can be determined within the tight-binding approximation [orlinear combination of atomic orbitals (LCAOs) approximation] as a sum ofatomic wave functions, B n , ku n0 ð ! rÞc X iC nl B n ð ! r ! ri Þ ð10Þwhere the sum is over lattice sites and n represents the conduction band orvalence band for the electron or hole, respectively, the nanocrystal problem isreduced to determining the envelope functions for the single-particle wavefunctions, f sp . Fortunately, this is exactly the problem that is addressed by theparticle-in-a-sphere model. For spherically shaped nanocrystals with a potentialbarrier that can be approximated as infinitely high, the envelopefunctions of the carriers are given by the particle-in-a-sphere solutions [Eq.(3)]. Therefore, each of the electron and hole levels depicted in Fig. 2b can bedescribed by an atomic-like orbital that is confined within the nanocrystal (1S,1P, 1D, 2S, etc.). The energy of these levels is described by Eq. (5), with thefree particle mass m 0 replaced by m c,v eff .Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.