Particle-hole excitonic effects

Crystals exhibit excitonic effects near the band edges, in which the Coulomb interaction between an electron and a valence band hole results in absorption which does not follow the one-particle joint density of states in Eq. (3.25). Excitons produce an absorption peak just below the band gap energy and modify the absorption at higher energies. There is no exciton absorption peak observable in any amorphous semiconductor, because it is broadened out by the disorder. The Coulomb interaction is present in a-Si H, but its significance in the optical absorption is unclear. 

Electron-hole correlations are an important aspect of the recombination. The recombination transition necessarily involves two particles, the electron and the hole, and so the recombination rate depends on whether they are spatially correlated or distributed randomly. The electron-hole pairing in a crystal is manifested in excitonic effects, but these are not detectable in the absorption spectra... 

Charge transfer excitations simulated by quasi-particle techniques lack the electron-hole interaction (excitonic effect) because the two excess charges... 

In this section we derive an effective Hamiltonian that describes the high energy physics associated with particle-hole (or ionic) excitations across the charge gap. The Hamiltonian will describe a hole in the lower Hubbard band and a particle in the upper Hubbard band, interacting with an attractive potential. This attractive potential leads to bound, excitonic states. In the next chapter we derive an effective-particle model for these excitons. A real-space representation of an ionic state is illustrated in Fig. 5.5(b). 

There is an important observation to be made about this effective-particle model. This is that since the exchange interaction, X, is local (i.e. it is only nonzero when r = 0), we immediately see that this term vanishes for odd parity excitons (namely, (r) = —i/ ni—r)), as tl>n 0) = 0. Now, since the parity of the exciton is determined by the particle-hole symmetry, and odd singlet and... 

Formally, the exciton binding energy is defined relative to the energy of a widely separated uncorrelated electron-hole pair. In practice, excitons whose particle-hole separation exceeds the length of the polymer (or more correctly, the conjugation length) can be considered unbound. This marks the breakdown of the effective-particle model. 

Table 8.1 shows that the dipole moment is largest for close lying exciton states, as the integral in eqn (8.31) is maximized when up — nq = 1. It also shows that the dipole moment increases as Up and n, increase, as the effective-particle wavefunction, -ipnir), spreads out (or the particle-hole separation increases) as n increases (as shown in Appendix E). 

There are also two other consequences of the electric field on the optical signature of an unbound particle-hole pair. First, the electroabsorption above the zero-field band gap exhibits oscillatory behaviour. This oscillatory behaviour can be traced to the oscillatory nature of the Airy functions, which are the solutions of the effective particle-hole equation in the absence of a Coulomb potential. Second, the position of the electroabsorption peaks vary as and not with the behaviour of excitons. 

The Franz-Keldysh effects (Weiser and Horvath 1997) have been successfully used to distinguish the particle-hole continuum from exciton states in polydiacetylene crystals (Sebastian and Weiser 1981). 

Fig. 36. Energy levels of excitonic states in CdS particles of various radii. Zero position of the lower edge of the conduction band in macrocrystalline CdS. Exc Energy of an exciton in macrocrystalline CdS. Effective masses of electrons and holes 0.19 m and 0.8 m respectively. The letters with a prime designate the quantum state of the hole...

Near k = 0 the electron and hole effective masses must be isotropic and constant. For a nanoparticle, the uncertainty in the exciton position depends upon its size. Ax 2R. If the relation between energy and momentum is independent of particle size the exciton... 

Cadmium sulfide suspensions are characterized by an absorption spectrum in the visible range. In the case of small particles, a quantum size effect (28-37) is observed due to the perturbation of the electronic structure of the semiconductor with the change in the particle size. For the CdS semiconductor, as the diameter of the particles approaches the excitonic diameter, its electronic properties start to change (28,33,34). This gives a widening of the forbidden band and therefore a blue shift in the absorption threshold as the size decreases. This phenomenon occurs as the cristallite size is comparable or below the excitonic diameter of 50-60 A (34). In a first approximation, a simple electron hole in a box model can quantify this blue shift with the size variation (28,34,37). Thus the absorption threshold is directly related to the average size of the particles in solution. 

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