Wannier exciton theory

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. 

The foundation of excitons theory was formulated by Frenkel, Peierls and Wannier (l)-(5) more than 70 years ago. After that time the theory has been enriched by many new aspects. The theory has also been exposed to continuous experimental verification, which has confirmed the role of excitons in such processes as absorption of light, luminescence and energy transfer, photochemical processes, etc. Before we examine the experiments, which illustrate the presence and the role of excitons in crystals, we will briefly describe the basic models of excitons, which are mostly used in the interpretation of experimental results. 

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 theory of this section relies on the assumption that excitons in light emitting polymers are Mott-Wannier excitons (as described in Chapter 6). The experimental and theoretical evidence for this assumption is described in detail in Chapter 11. 

We evaluate this matrix element using the effective-particle exciton model introduced in Chapter 6. We briefly review this theory here. In the weak-couphng limit (namely, the limit that the Coulomb interactions are less than or equal to the band width) the intramolecular excited states of semiconducting conjugated polymers are Mott-Wannier excitons described by,... 

The bandlike aspects of the excitonic insulator model can be replaced by a more localized description developed from liquid-state theory (Hall and Wolynes, 1986 Lx>gan and Edwards, 1986 Xu and Stratt, 1989). In essence, the delocalized Wannier) exciton is replaced by a localized Frenkel) exciton. Such calculations exhibit a sharp transition in the degree of hybridization as the density of the system is varied. When the mercury density reaches a critical value, the degree of -character in the ground state drops sharply from the 100% (pure s-) atomic value. In common with other exciton models, the electronic transitions associated with electric dipole interactions in the localized limit are strongly enhanced by clustering. 

For the theoretical description of the optical properties In covalently bound macromolecules neither of the classical (Frenkel- or Wannier-type) exciton models is appropriate. A physically more sound basis for the solution of this problem is provided by Takeuti s charge transfer exciton theory (15) within the framework of the... 

The EMA is a quasi-particle theory, which treats the hole created in the valence band and electron excited to the conduction band, as free particles whose effective masses are determined by a quadratic fit to the curvature at the band minima (maxima) of the conduction (valence) band (Fig. 1). If we add the coulumbic attraction of an electron and hole to this picture, we can have a theoretically simple manifestation of an exciton. The electron and hole are bound together by a screened coulomb interaction to form a so-called Mott-Wannier exciton [29],This exciton presents an energy spectrum analogous to a hydrogen atom (i.e. with radial and angular quantum number) but it is further complicated by the fact that it is coupled to a thermal bath of phonons and that the mass of an exciton is energy dependent. 

The theory of Wannier-Mott excitons and, in particular, the limits of validity of eqn (1.3), can be found, for instance, in the textbook by Knox (8) and the review article by Haken (19). 

A similar situation appears in the theory of Wannier-Mott excitons (see, for example, (17), 4), where the interaction between the electron and the hole is given by —e2/er only at large enough distances. 

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