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Figure 2.20. First few energy levels in the Rydberg series of a hydrogen atom (left), positronium (center), and a typical exciton (right).

Figure 2.20. First few energy levels in the Rydberg series of a hydrogen atom (left), positronium (center), and a typical exciton (right).

Table B.ll that the relative dielectric constant e/e0 has the range of values 7.2 < e/Sq < 17.7 for these materials, where e0 is the dielectric constant of free space. Both of these factors have the effect of decreasing the exciton energy Eex from that of positronium, and as a result this energy is given by m-Ano ^ 13-6^ev (2.18)

as shown plotted in Fig. 2.20. These same two factors also increase the effective Bohr radius of the electron orbit, and it becomes

Using the GaAs electron effective mass and the heavy-hole effective mass values from Table B.8, Eq. (2.17) gives m*/m0 = 0.059. Utilizing the dielectric constant value from Table B.ll, we obtain, with the aid of Eqs. (2.18) and (2.19), for GaAs

where E0 is the ground-state (n = 1) energy. This demonstrates that an exciton extends over quite a few atoms of the lattice, and its radius in GaAs is comparable with the dimensions of a typical nanostructure. An exciton has the properties of a particle; it is mobile and able to move around the lattice. It also exhibits characteristic optical spectra. Figure 2.20 plots the energy levels for an exciton with the ground-state energy £0 = 18 meV.

Technically speaking, the exciton that we have just discussed is a weakly bound electron-hole pair called a Mott-Wannier exciton. A strongly or tightly bound exciton, called a Frenkel exciton, is similar to a long-lived excited state of an atom or a molecule. It is also mobile, and can move around the lattice by the transfer of the excitation or excited-state charge between adjacent atoms or molecules. Almost all the excitons encountered in semiconductors and in nanostructures are of the Mott-Wannier type, so they are the only ones discussed in this book.

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