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

energy splitting and the transition oscillator strengths of the split-off states, aswell as their order, are very sensitive to the NC size and shape, as shown insubsections B and C. We calculate this splitting neglecting the warping of thevalence band and the nonparabolicity of the electron and light-hole energyspectra.B. Energy Spectrum and Wave FunctionsNanocrystal asymmetry lifts the hole state degeneracy. The asymmetry hastwo sources: the intrinsic asymmetry of the hexagonal lattice structure of thecrystal [12] and the non-spherical shape of the finite crystal [14]. Both split thefourfold degenerate hole state into two twofold degenerate states—a Kramer’sdoublet—having jMj = 1/2 and 3/2, respectively.The splitting due to the intrinsic hexagonal lattice structure, D int , can bewritten [12]D int ¼ D cr vðbÞ;ð8Þwhere D cr is the crystal field splitting equal to the distance between the A and Bvalence subbands in bulk semiconductors having a hexagonal lattice structure(25 meV in CdSe). Equation (8) is obtained within the framework of thequasicubic model for the case when the crystal field splitting can be consideredas a perturbation [12]. The Kramer’s doublet splitting does not depend onthe NC size but only on the ratio of the light- to heavy-hole effective masses.The dimensionless function v(b) [12] that describes this dependence (shown inFig. 1b) varies rapidly in the region 0 < b < 0.3. The jMj = 3/2 state is theground state.We model the nonsphericity of a NC by assuming that it has anellipsoidal shape. The deviation from the sphericity is quantitatively characterizedby the ratio c/b = 1 + l of the ellipsoid’s major (c) to minor (b) axes,where l is the NC ellipticity, which is positive for prolate particles andnegative for oblate particles. The splitting arising from nonsphericity can becalculated in the first-order perturbation theory [14], which yieldsD sh ¼ 2luðbÞE 3=2 ðbÞð9Þwhere E 3/2 is the 1S 3/2 ground-state hole energy for spherical NCs of radiusa = (b 2 c) 1/3 . E 3/2 is inversely proportional to a 2 [see Eq. (2)] and the shapesplitting is therefore a sensitive function of the NC size. The function u(b) [14]is equal to 4/15 at b= 0. It changes sign at b = 0.14, passes a minimum at b c0.3, and, finally, becomes 0 at b = 1 (see Fig. 1c).The net splitting of the hole state, D(a, b, l), is the sum of the crystalfield and shape splitting:Dða; b; lÞ ¼ D sh þ D intð10ÞCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.