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

In large NCs, the magnetic field splitting l B g e H is of the same order asthe exchange interaction g and cannot be considered as a perturbation. At thesame time, both of these energies are much smaller than the splitting due tothe crystal asymmetry. We consider here the admixture in the jFj=2 darkexciton of the lowest jFj=1 exciton only. This problem can be calculatedexactly. The magnetic field also lifts the degeneracy of the exciton states withrespect to the sign of the total angular momentum projection F. The energiesof the former F = 2 and F = 1 states aree ¼ 1; 2 ¼ D þ 3l Bg h H z2Fqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3g þ l B g e H z Þ 2 þ ðl B g e Þ 2 H?22ð47Þwhere + ( ) refers to the F = 1 state with an qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF = 2 admixture ( F = 2state with an F = 1 admixture) and H ? ¼ Hx 2 þ H y2 . The correspondingwave functions arepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1=2C F 1; 2 ¼ 2 þ j n j 2 FppffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2 p 2 þ j n j 2 z; 3=2ð48Þnb h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1=2C #; 3=2 ;2 p 2 þ j n j 2 p 2 þ j n j 2 Fpwhere n = l B g e H + and p = 3g + l B g e H z . The energies and wave functionsof the former F = 2, 1 states are (using notation similar to that used justabove)e F 1;2 ¼C F 1;2 ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD þ 3l Bg h H zð3g l B g e H z Þ 2 þ ðl B g e Þ 2 H 2F22pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1=2 V2 þ n V2 FpVpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC #;3=22 p V2 þ n V2n Vð49Þb h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1=2C z;3=2 ; ð50Þ2 p V2 þ n V2 p V2 þ n V2 Fp Vwhere nV = l B g e H and pV = 3g l B g e H z . As a result, the decay time of thedark exciton in an external magnetic field can be written [3]1s r ðH; xÞ ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ f 2 þ 2fxp2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ f 2 þ 2fx1 fx32s 0ð51ÞCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.