States and Excitons

01 eV most donor-impurity states are empty -the donor is ionized and the electron is free. 

In the semiconductors of greater polarity, the dielectric constants are smaller and the effective masses larger, and the same evaluation leads to 0.07 eV in zinc selcnidc, for example many of the impurity states can be occupied at room temperature. As the energy of the impurity states becomes deeper, the effective Bohr radius becomes smaller and the use of the effective mass approximation becomes suspect the error leads to an underestimation of the binding energy. Thus, in semiconductors of greatest polarity- and certainly in ionic crystals— impurity states can become very important and arc then best understood in atomic terms. We will return to this topic in Chapter 14, in the discussion of ionic crystals. 

I irsi evaluate the iieare.st-iicighbor niatrix elements V,. using die bond length of 1,42 A (.see Fig.. 4-10.) Wc neglect second-neighbor niatrix elements and matrix elements between successive graphite planes. 

There arc two atoms per primitive cell, so two Bloch sums of the form ofEq. (6-7) are needed (with a bond orbital replaced by p orbitals oriented perpendicular to the graphite plane). The expectation value of the Hamiltonian with respect to cither of the Bloch sums taken individually is simply Cp. The matrix element between the two is of a form analogous to Eq. (6-9), but with only a single type of nearest-neighbor matrix element and with three rather than two terms. They may be written for arbitrary k in terms of the three nearc.st-neighbor vectors dj, da, and da.  

Obtain the bands explicitly and plot them for k along a nearest-neighbor vector, d. This vector is in the direction of a zone edge of the hexagonal Brillouin Zone, reaching the edge at /f = 27c/(3d). The results may be compared with the n bonds of Painter and Ellis (1970), shown in Problem 3-3. 

---

In the case of benzene (point group D6h) excimer states arise from configuration interaction of the 8-fold degenerate CR states and exciton states of both 1La and 1Lb origin. Owing to the very large separation of 1La and 1Lb states in this molecule, the lowest exciton state is of 1Lb character and contributes to the lowest excimer state after configuration interaction.68... 

Another interesting example concerning exciton dynamics was also described by Kim and Weissman (83,84). They applied the transient magnetization technique to the study of the photoexcitation of phenazine doped with anthracene. In this system photo-excitation in the phenazine absorption region leads to a selectively populated phenazine triplet exciton which transfers the excitation to anthracene with conservation of polarization (27). The authors emphasized that the time dependence of the transient response requires further analysis, which must include an adequate quantum theoretical treatment on the evolution of the system shortly after excitation. One can expect that future refinement of this technique will lead to more exciting studies of the dynamics of triplet states and excitons in solid state. 

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. 

Exciton States and Exciton Transport in Molecular Crystals... 

The partial yield of electrons in an energy window AE at a fixed final state energy E, as a function of photon energy, where E is fixed at < 5eV so that only secondary electrons are measured, is referred to as partial yield spectroscopy. When E > 5eV, although the technique is experimentally identical, it is used to study initial state and excitonic effects and is known as constant final state spectroscopy. 

Glaeske H., Malyshev V.A., Feller K-H. (2001). Mirrorless optical bistability of an ultrathin glassy film built up of oriented J-aggregates Effects of two-exciton states and exciton-exciton annihilation, /, Chem. Phys. Vol.114. pp.1966-1969... 

Riter R E, Edington M D and Beck W F 1997 Isolated-chromophore and exciton-state photophysics in C-phycocyanin trimers J. Phys. Chem. B 101 2366-71... 

Edington M D, RIter R E and Beck W F 1996 Interexciton-state relaxation and exciton localization In allophycocyanin trimers J. Phys. Chem. 100 14 206-17... 

Wlrile tire Bms fonnula can be used to locate tire spectral position of tire excitonic state, tliere is no equivalent a priori description of the spectral widtli of tliis state. These bandwidtlis have been attributed to a combination of effects, including inlromogeneous broadening arising from size dispersion, optical dephasing from exciton-surface and exciton-phonon scattering, and fast lifetimes resulting from surface localization 1167, 168, 170, 1711. Due to tire complex nature of tliese line shapes, tliere have been few quantitative calculations of absorjDtion spectra. This situation is in contrast witli tliat of metal nanoparticles, where a more quantitative level of prediction is possible. 

These two types of exciton are schematically illustrated in Figure 4.13. The Mott-Wannier excitons have a large radius in comparison to the interatomic distances (Figure 4.13(a)) and so they correspond to delocalized states. These excitons can move freely throughout the crystal. On the other hand, the Frenkel excitons are localized in the vicinity of an atomic site, and have a much smaller radius than the Mott-Wannier excitons. We will now describe the main characteristics of these two types of exciton separately. 

---