Excitons effective-particle model

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). 

Fig. 6.2. The effective-particle model of excitons on a linear chain. The total exciton wavefunction, = ipn r) j R), where ipn r) is the relative wavefunction...

Notice that two quantum numbers specify the exciton eigenstates, eqn (6.13) or eqn (6.16) the principle quantum number, n, and the (pseudo) momentum quantum number, K (or fUj). For every n there are a family of excitons with different centre-of-mass momenta, and hence different centre-of-mass kinetic energy. Odd and even values of n correspond to the relative wavefunction, tl>n r), being even or odd under a reversal of the relative coordinate, respectively. We refer to even and odd parity excitons as excitons whose relative wavefunction is even or odd imder a reversal of the relative coordinate. This does not mean that the overall parity of the eigenstate (eqn (6.12)), determined by both the centre-of-mass and relative wavefiictions, is even or odd. The number of nodes in the exciton wavefunction, V rt( ), is n— 1. Figure 6.2 illustrates the wavefunctions and energies of excitons in the effective-particle model. 

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... 

In this chapter we have described the effective-particle models for excitons in the weak and strong coupling limits, and compared them to essential exact, numerical (DMRG) calculations. We saw that there is good agreement between the effective-particle models and the computational results in these limits. We used these extreme limits to understand the numerical calculations in the intermediatecoupling regime. We summarize the main points as follows ... 

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. 

As described in Chapter 6, n = 1 corresponds to the Sx and T% families of intrachain excitons, while n = 2 corresponds to the ct and Tct families of intrachain excitons. The lowest energy branch of each family has the smallest pseudo-momentum, namely, j =. The effective-particle model is illustrated in Fig. 6.2. 

The Schodinger equation for the effective-particle model of excitons was introduced in Chapter 6. In this appendix we derive that equation. 

Electronic dipole moments Application of the exciton model Flaving discussed the vibrational overlaps, the final task is to evaluate the electronic transition dipole moments. We obtain insight into the behaviour of the dipole moments by using the effective-particle exciton model, introduced in Chapter 6. 

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,... 

In this appendix we examine the properties of the effective-particle exciton models derived in Appendix D and described in Chapter 6 in the continuum or effective-mass limit. 

In this appendix we use the effective-particle exciton models introduced in Chapter 6 to calculate transition dipole moments. These results are summarized in Chapter 8. 

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. 

Absorption of the X-ray makes two particles in the solid the hole in the core level and the extra electron in the conduction band. After they are created, the hole and the electron can interact with each other, which is an exciton process. Many-body corrections to the one-electron picture, including relaxation of the valence electrons in response to the core-hole and excited-electron-core-hole interaction, alter the one-electron picture and play a role in some parts of the absorption spectrum. Mahan (179-181) has predicted enhanced absorption to occur over and above that of the one-electron theory near an edge on the basis of core-hole-electron interaction. Contributions of many-body effects are usually invoked in case unambiguous discrepancies between experiment and the one-electron model theory cannot be explained otherwise. Final-state effects may considerably alter the position and strength of features associated with the band structure. 

Many models have been presented to explain quantitatively the dependence of exciton energy on the cluster size [7, 11, 21-30]. This problem was first treated by Efros et al. [7], who considered a simple particle in a box model. This model assumes that the energy band is parabolic in shape, equivalent to the so-called effective mass approximation. The shift in absorption threshold, AE, is dependent upon the value of the cluster radius, R, Bohr radius of the electron, ae ( = h2v.jmce2), and Bohr radius of the hole, ah (= h2e/mhe2). When (1) R ah and R ae, and (2) ah R ac,... 

The optical bleaching by stored electrons is the basis for the occurrence of strong optical nonlinearity observed in Q-particles [64]. The physical reason for this optical bleaching is still not discussed conclusively in literature. The most obvious explanation comes from a state filling model. The stored electrons occupy the lowest electronic levels in the conduction band and, consequently, the optical transition has to occur to higher electronic levels (i.e., at shorter wavelength). This effect is known in solid-state semiconductor physics as the Burstein shift [65]. Other theoretical models describe the optical bleaching as a consequence of the polarization of the exciton in the electric field of the stored electron, which is then... 

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