Excitons particle-hole separation

Weak electron-electron screening (the dielectric constant in the solid state is e 2 - 3). The lowest energy excitons have large binding energies ( 0.5 — 1.5 eV) and small particle-hole separations ( 5 — 10 A). 6, 10, 11... 

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. 

The particle-hole separations are shown in Fig. 6.6 at 102 sites. The jumps in the separation occur at p = 9 and p = 8 for the even and odd parity excitons, respectively, corresponding to the j = 1 branches of the n = 3 and n = 4 families of excitons. Notice that, as predicted in Appendix E, the particle-hole separations decrease with increasing j for the same n. 

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). 

Mott-Wannier excitons were described in Chapter 6. Recall that our definition of Mott-Wannier excitons includes bound particle-hole excitations with small particle-hole separations. 

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... 

We know that, in semiconductors or insulators, valence and conduction bands are separated by some finite energy gap characteristic of the material. When an electron from the valence band gets sufficient energy to overcome the energy gap may be by thermal excitation or absorption of photons, and it goes to conduction band, a hole is left behind. The electron-hole pair so formed is a quasi-particle called exciton. An exciton can move in the crystal whose centre of mass motion is quantized. Different kinds of excitons can be identified in a variety of materials. If the electron-hole bound pair is tightly-bound with distance of electron-hole pair comparable to lattice constant, then it is called Frenkel exciton. On the other hand, one may have an exciton with electron-hole separation... 

The primary events occurring within a nanometer-sized semiconductor particle after the absorption of a photon the energy of which is exceeding the bandgap energy have been discussed in detail based upon a review of the current literature. Both, electrons and holes, are separated extremely rapidly from the initially formed exciton and trapped at or very close to... 

If f i a,B (weak confinement), the splitting between the levels in a spherical well is much smaller than the binding energy of a three-dimensional Wannier-Mott exciton. In this case the latter can be assumed to be a rigid particle moving in a spherical well. The variables corresponding to the relative motion of an electron and a hole and the motion of the center of mass effectively separate, and the wavefunction factorizes. Then... 

Quantum confinement is defined as the space where the motions of electrons and holes in a semiconductor are restricted in one or more dimensions. This quantum confinement occurs when the size of semiconductor crystallites is smaller than the bulk exciton Bohr radius. Quantum wells, quantum wires, and quantum dots are confined in one, two, and three dimensions, respectively [1, 2]. The confinement can be created due to electrostatic potentials, the presence of an interface between different semiconductor materials, and the presence of a semiconductor surface. A valence band and a conduction band are separated by an energy range known as the band gap ( g). These amounts of energy will be absorbed in order to promote an electron from the valence band to the conduction band and emitted when the electron relaxes directly fi om the conduction band back to the valence band. By changing the size of the semiconductor nanoparticles, the energy width of the band gap can be altered and the optical and electrical responses of these particles are changed (Fig. 1). 

Since the excited states are electrically neutral, only the short-range, acoustic interaction is relevant in eqn (7.24). (This is also true for the exciton-polaron, as the particle and hole are closely separated.) The polaron, however, being charged also couples to the longitudinal optic phonons, so the long-range term is retained... 

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