Wannier-Mott exciton

Figure 4.13 The schemes of (a) a weakly bound (Mott-Wannier) exciton and (b) a tightly bound (Frenkel) exciton.

These two types of exciton are schematically illustrated in Figure 4.13. The Mott-Wannier excitons have a large radius in comparison to the interatomic distances (Figure 4.13(a)) and so they correspond to delocalized states. These excitons can move freely throughout the crystal. On the other hand, the Frenkel excitons are localized in the vicinity of an atomic site, and have a much smaller radius than the Mott-Wannier excitons. We will now describe the main characteristics of these two types of exciton separately. 

Thus, Mott-Wannier excitons can give rise to a number of absorption peaks in the pre-edge spectral region according to the different states = 1, 2, 3,... As a relevant example. Figure 4.14 shows the low-temperature absorption spectrum of cuprous oxide, CU2O, where some of those hydrogen-like peaks of the excitons are clearly observed. These peaks correspond to different excitons states denoted by the quantum numbers = 2, 3, 4, and 5. 

Crystal (Mott-Wannier excitons) Eg (eV) Eg (meV) Crystal (Frenkel excitons) Eg (eV) Ei (meV)... 

For weakly bound (Mott-Wannier) excitons (mainly observed in semiconductors), the binding energies are in the meV range, as can be appreciated from Table 4.4. Inspection of this table also shows a general trend Ei, tends to increase as increases. 

In tightly bound (Frenkel) excitons, the observed peaks do not respond to the hy-drogenic equation (4.39), because the excitation is localized in the close proximity of a single atom. Thus, the exciton radius is comparable to the interatomic spacing and, consequently, we cannot consider a continuous medium with a relative dielectric constant as we did in the case of Mott-Wannier excitons. 

Mobile defects are Frenkel78 excitons, Mott-Wannier excitons, polarons, bipolarons, polaritons, and solitons. 

A Mott-Wannier exciton is a neutral quasi-particle, consisting of an excited bound-state electron and its associated "Coulomb hole" in a high-dielectric constant solid, that can also travel throughout the lattice without transporting net charge since the exciton radius is several lattice constants, its binding energy is as low as 0.01 eV it thus tends to be more "delocalized" than the Frenkel exciton. 

The spectra may also be described in the language of solid state theory. The atomic excited states are the same as the excitons that were described, for semiconductors, at the close of Chapter 6. They are electrons in the conduction band that are bound to the valence-band hole thus they form an excitation that cannot carry current. The difference between atomic excited states and excitons is merely that of different extremes the weakly bound exciton found in the semiconductor is frequently called a Mott-Wannier exciton-, the tightly bound cxciton found in the inert-gas solid is called a Frenkel exciton. The important point is that thecxcitonic absorption that is so prominent in the spectra for inert-gas solids does not produce free carriers and therefore it docs not give a measure of the band gap but of a smaller energy. Values for the exciton energy are given in Table 12-4. 

Inorganic semiconductors and metals, on the other hand, consist of a network or ordered atoms with no discernible molecular unit. Strong chemical bonds, rather than van der Waals forces, exist between atoms. For a semiconductor crystal, electronic excitation consists of a loosely bounded electron-hole pair (the Mott-Wannier exciton [19]), usually delocalized over a length much longer than the lattice constant. The Bohr radius of the exciton is given by... 

Here, /Xexc is the reduced effective mass of the exciton, in terms of the free electron mass, and e is the dielectric constant. In addition, there are also Frenkel excitons, which are spatially more localized than Mott-Wannier excitons. They cannot be represented by a series formula. Therefore, Frenkel excitons have been assigned to single low-lying absorption bands in the near-edge regime of solid rare gases. 

The study of excitons in conjugated polymers has often been inspired by the treatment of excitons in bulk three-dimensional semiconductors (as described in Knox (1963)). A particle-hole excitation from the valence band to the conduction band in a semiconductor leaves a positively charged hole in the valence band and a negatively charged electron in the conduction band. The Coulomb attraction between these particles results in bound states, or excitons. In three-dimensional semiconductors the excitons are usually weakly bound, with large particle-hole separations, and are well described by a hydrogenic model. Excitons in this limit are known as Mott- Wannier excitons. 

For simplicity, however, we prefer to denote all excitons formed from bound states of conduction band electrons and valence band holes as Mott-Wannier excitons, recognizing that this term includes both small and large radius excitons. We call this limit the weak-coupling limit, as the starting point in the construction of the exciton basis is the noninteracting band limit. As we will see, a real space description of a Mott-Wannier exciton is of a hole in a valence band Wannier orbital bound to an electron in a conduction band Wannier orbital. 

Fig. 6.1. The real-space particle-hole excitation, R- -r/2, R—r/2), labelled 1, from the valence band Wannier orbital at R — r/2 to the conduction band valence orbital at R+r/2. Its degenerate counterpart, R—r/2, R+r/2), connected by the particle-hole transformation, is labelled 2. R = (ve + rh)/2 is the centre-of-mass coordinate and r = (re — Vh) is the relative coordinate. A Mott-Wannier exciton is a bound particle-hole pair in this representation.

Table 6.1 The classification of the many body singlet exciton states with particle-hole symmetry in terms of their Mott- Wannier exciton quantum numbers (The corresponding triplet states with the same spatial symmetry but opposite particle-hole symmetry have the same quantum numbers)...

Wannier triplet exciton to a gapless spin-density-wave (or magnon), while the 2 A+ state has evolved from the weak-coupling n = 2, j = Mott-Wannier exciton to a pair of triplets (or a bimagnon) (Schulten and Karplus 1972 Tavan and Schulten 1987). This picture is confirmed by the numerical calculations for six sites, presented in Table 6.4. The first odd parity singlet exciton is now the m A+ state, where m > 2 (m = 5 for the six-site calculation). This, as predicted, is virtually degenerate with its associated even parity exciton, the state,... 

At = 0.1 there are both Mott-Hubbard and Mott-Wannier excitons, forming two inter-related families of essential states. In general, the B states are linear superpositions of eqns (6.22) and (6.30), while the states are linear superpositions of eqns (6.23) and (6.31). As the bond dimerization decreases the spin-density-wave component of the state increases (Mukhopadhyay et al. 1995). Figure 6.10(c) shows the l H, 2 A+, and 4 H states, predominately forming the Mott-Wannier family of excitons, while Fig. 6.10(d) shows the l B, ... 

In the intermediate-coupling regime Mott-Wannier excitons are the more appropriate description for large dimerization (J = 0.2), while for the undimerized chain Mott-Hubbard excitons are the correct description. For dimerizations relevant to polyacetylene and polydiacetylene (that is, S ... 

The effects of electron-phonon interactions alone were described in Chapter 4. We showed that these interactions lead to a dimerized, semiconducting ground state and to solitonic structures in the excited states. On the other hand, the effects of electron-electron interactions in a polymer with a fixed geometry were described in Chapters 5 and 6. There it was shown that the electronic interactions cause a metal-insulator (or Mott-Hubbard) transition in undimerized chains. Electron-electron interactions also cause Mott-Wannier excitons in the weak-coupling limit of dimerized chains, and to both Mott-Hubbard excitons and spin density wave excitations in the strong coupling limit. 

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