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

square of the matrix element of the momentum operator epˆ between this stateand the vacuum stateP F ¼ jh0jepj ˆ C ˜ Fij 2ð24Þwhere j0i = d(r e r h ) and e is the polarization vector of the emitted orabsorbed light. The momentum operator pˆ acts only on the valence-bandBloch functions [see Eq. (5)], and the exciton wave function, ˜C F , is written inthe electron–electron representation. Exciton wave functions in the electron–hole representation are transformed to the electron–electron representationby taking the complex conjugate of Eqs. (17), (19), and (23) and flippingthe spin projections in the hole Bloch functions (z and # to # and z,respectively).To calculate the matrix element for a linear polarized light, we expandthe scalar product epˆ asepˆ¼ e z pˆz þ 1 2 ½epˆþ þ epˆŠð25Þwhere z is the direction of the hexagonal axis of the NC, e F = e x F ie y , pˆ F =pˆx F ipˆy, and e x,y and pˆx,y are the components of the polarization vector andthe momentum operator, respectively, that are perpendicular to the NC hexagonalaxis.Using this expansion in Eq. (24), one can obtain for the exciton statewith F = 0 [3]:P U;L0¼ jh0jepj ˆ˜CU;L0 ij 2 ¼ N U;L0cos 2 ðh lp Þ ð26Þwhere N L 0 = 0, N U 0 = 4KP 2 /3, P = hSjpˆzjZi is the Kane interband matrixelement, h lp is the angle between the polarization vector of the emitted orabsorbed light and the hexagonal axis of the crystal, and K is the square of theoverlap integral [12]:K ¼ 2 Za dr r sin pr R 0 ðrÞj 2að27ÞThe magnitude of K depends only on b and is independent of the NC size;hence, the excitation probability of the F = 0 state is also size independent.For the lower exciton state, 0 L , the transition probability is proportional toN 0 L and is identically zero. At the same time, the exchange interactionincreases the transition probability for the upper 0 U exciton state (it isproportional to N 0 U ) by a factor of 2. This result arises from the constructiveand destructive interference of the wave functions of the two indistinguishableexciton states jz, 1/2i and j#, 1/2i [see Eq. (23)].Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.