## Excitons

Excitons, which were introduced in Section 2.3.3, are a common occurrence in semiconductors. When an atom at a lattice site loses an electron, the atom acquires a positive charge that is called a hole. If the hole remains localized at the lattice site, and the detached negative electron remains in its neighborhood, it will be attracted to the positively charged hole through the Coulomb interaction, and can become bound to form a hydrogen-type atom. Technically speaking, this is called a Mott-Wannier type of exciton. The Coulomb force of attraction between two charges Qt = — e and Qh = +e separated by a distance r is given by F = —ke^/sr2, where e is the electronic charge, k is a universal constant, and £ is the dielectric constant of the medium. The exciton has a Rydberg series of energies E sketched in Fig. 2.20 and a radius given by Eq. (2.19): atS = 0.0529(e/e0)/(m*/w0), where k/eq is the ratio of the dielectric constant of the medium to that of free space, and m*/m0 is the ratio of the effective mass of the exciton to that of a free electron. Using the dielectric constant and electron effective mass values from Tables B.l 1 and B.8, respectively, we obtain for GaAs

showing that the exciton has a radius comparable to the dimensions of a typical nanostructure.

The exciton radius can be taken as an index of the extent of confinement experienced by a nanoparticle. Two limiting regions of confinement can be identified on the basis of the ratio of the dimension d of the nanoparticle to the exciton radius Ojjf, namely, the weak-confinement regime with d > a^ (but not d a^) and the strong-confinement regime d < a^. The more extended limit d » aefr corresponds to no confinement. Under weak-confinement conditions the exciton can undergo unrestricted translational motion, just as in the bulk material, but for strong confinement this translation motion becomes restricted. There is an increase in the spatial overlap of die electron and hole wavefunctions with decreasing particle size, and this has die effect of enhancing the electron-hole interaction. As a result, the energy splitting becomes greater between the radiative and nonradiative exciton states. An optical index of die confinement is the blue shift (shift to higher energies) of the optical absorption edge and the exciton energy with decreasing nanoparticle size. Another result of the confinement is die appearance at room temperature of excitonic features in the absorption spectra that are observed only at low temperatures in the bulk material. Further details on exciton spectra are provided in Sections 2.3.3 and 8.2.1.

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