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

where n h em is the hole ‘‘emitting’’ state/manifold occupation number. The holecontribution to TA is strongly dependent on the degeneracy of the ‘‘emitting’’hole state. In the case of the twofold degeneracy, the sum of average electronand hole occupation numbers can be presented as hn e 1S þ nh em i ¼ 2 1 e hN i 1 þ hNi 2ð7ÞAt the gain threshold, Da em ( tx)/a 0 ( tx) = 1 and, hence [see Eq. (3)],hn e 1S þ nh em i th ¼ 1ð8ÞUsing Eq. (8), we determine that the gain threshold corresponds to thecondition hNi th = 1.15, which is close to the theoretical one e–h pairthreshold obtained without statistical averaging of NQD occupation numbers[2].The results of effective mass calculations [48], as well as measurementsof temperature-dependent radiative lifetimes [49], suggest that at temperaturesabove f50 K, the lowest ‘‘emitting’’ NQD state can be described interms of a strong mixing of the lowest ‘‘dark’’ exciton (total momentumprojection J = F2) and the next bright-exciton state with J = F1. This resultsuggests that the lowest ‘‘emitting’’ hole state can be characterized by afourfold degeneracy, and its average population number can be calculated ashn h em i = 1 0.25 S4 i¼1 iP(i). Combining the latter expression with Eqs. (5) and(8), we obtain hNi th = 1.55. As expected, the gain threshold in the case of thefourfold degeneracy of the ‘‘emitting’’ hole state is higher than in the case ofthe twofold degeneracy.The fact that optical gain develops at hNi>1 indicates that it arises fromNQDs which contain multiple e–h pairs (i.e., NQD multiexciton states).Because of the Auger recombination, the intrinsic lifetime of multiexcitonstates in NQDs rapidly decreases as the number of e–h pairs per dot isincreased (see Sect. IV.B), suggesting that optical gain is primarily due to twopairstates (biexcitons) which have the longest relaxation time among thestates with N z 2 [50].Optical-gain pump dependences and gain thresholds were studiedexperimentally in Ref. 50 using matrix-free films of CdSe NQDs. Figure 16displays room-temperature emission and linear (a 0 ) and nonlinear (a) absorptionspectra of TOPO-capped CdSe NQDs with R = 2.5 nm. Thenonlinear absorption/gain and emission spectra were taken at a pump fluenceW = 1 mJ/cm, which was above both the gain threshold and the threshold forthe development of amplified spontaneous emission (ASE) observed as aCopyright 2004 by Marcel Dekker, Inc. All Rights Reserved.