Weakly Bound Mott-Wannier Excitons

The simplest exciton can be modeled by a hydrogen atom, with the electron and hole (equivalent to the hydrogen nucleus) in a stable orbit around each other. Within this model, two basic types of excitons can occur in crystalline materials  

Within the simple Bohr model used for weakly bound excitons, the radius of the electron-hole orbit is given by 

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. 

This is mainly because By decreases and /x increases as the band gap increases. 

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

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. 

Optical excitations in molecular crystals are well known as Frenkel excitons and the detailed descriptions have been derived by Davydov [4] and Craig and Walmsley [5]. Molecular excitons resemble very much the optical properties of the isolated molecules, since the exciton is confined on one molecule and only the weak interaction with the surrounding molecules leads to the formation of a collective excitation. This is contrary to the large radius Mott-Wannier excitons in conventional semiconductors, where the electron and the hole are typically loosely bound with... 

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