Mott-Hubbard excitons

Mott-Wannier and Mott-Hubbard excitons are described in the next chapter. 

Fig. 6.8. A Mott-Hubbard exciton. (a) An empty orbital (holon) at i — ijl is bound to a doubly occupied orbital (doublon) at i - -ij2. This is equivalent to two spinless fermions, or hardcore bosons, represented by Xs, shown in (b).

We can see this behaviour by stud3dng the numerical calculations. First, consider (i = 0. Figure 6.10(a) shows the four lowest essential states. The 9 A+, and states are the j = 1, n = 1,2, and 3 Mott-Hubbard excitons. 

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 ... 

It is instructive to apply these exciton theories to actual conjugated polymers. Calculations on single poly(para-phenylene) chains (see Section 11.2.3) predict the l Sj (n = 1, J = 1) exciton at 3.7 eV, the (n = 2, j = 1) exciton at 5.1 eV and the l A triplet close in energy to the 2 A+ state, at 5.5 eV. This progression indicates a Mott-Wannier series of excitons. An equivalent description applies to poly(para-phenylene-vinylene). In contrast, polyacetylene and polydiacetylene have predominately Mott-Hubbard excitons. In polyacetylene the vertical energies of the and 2M+ states are virtually degenerate... 

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. 

Figure 10.2 illustrates the electroabsorption spectrum of phenyl-substituted traras-polyacetylene thin film (Liess et al. 1997). The feature at 2.0 eV is the red-shifted l Bu exciton. The feature at 2.5 eV is attributed to a dipole-forbidden state, namely the m Ag state. Unlike polydiacetylene crystals, disordered trans-polyacetylene thin film does not exhibit Pranz-Keldysh oscillations (described in Chapter 8) and therefore a definite assignment of a conduction band edge cannot made. However, because disordered polydiacetylene also does not exhibit Pranz-Keldysh oscillations, but a smeared-out feature similar to the one exhibited at 2.5 eV in Fig. 10.2 it is sometimes assumed that this feature does mark the band edge. Another interpretation is that this feature represents the n = 2 Mott-Hubbard exciton, described in Chapter 6, with the particle-hole continuum lying close in energy (possibly at 2.7 eV, which is three times the THG feature at... 

This result can also be derived by noting that Mott-Hubbard excitons include some ground state (or covalent) character, with amplitudes 0 t/U). 

An opposite, strong-coupling limit has also been used to describe excitons in conjugated polymers (Gallagher and Mazumdar 1997 Gebhard et al. 1997 Essler et al. 2001 Harford 2002). As described in the previous chapter, in this limit a correlation gap separates the electron removal spectral weight (the lower Hubbard band) from the electron addition spectral weight (the upper Hubbard band). Now the bound particle-hole excitations are Mott-Huhhard excitons. That is, a particle excited from the lower Hubbard band to the upper Hubbard band... 

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