Excitons

From the preceding derivations we conclude that the dielectric function of a semiconductor or insulator will derive from interband contributions only, as given by Eqs. (5.54) and (5.55) while that of a metal with several bands will have interband contributions as well as intraband contributions, described by Eqs. (5.51) and (5.52) (see also Ref. [61]). A typical example for a multi-band rf-electron metal, Ag, is shown in Fig. 5.4. For more details the reader is referred to the review articles and books mentioned in the Further reading section. For semiconductors, it is often easy to identify the features of the band structure which are responsible for the major features of the dielectric function. The latter are typically related to transitions between occupied and unoccupied bands which happen to be parallel, that is, they have a constant energy difference over a large portion of the BZ, because this produces a large joint density of states (see problem 8). 

Up to this point we have been discussing excitations of electrons from an occupied to an unoccupied state. We have developed the tools to calculate the response of solids to this kind of external perturbation, which can apply to any situation. Often we are interested in applying these tools to situations where there is a gap between occupied and unoccupied states, as in semiconductors and insulators, in which case the difference in energy between the initial and final states must be larger than or equal to the band gap. This implies a lower cutoff in the energy of the photons that 

Binding energies of the excitons are given in units of electronvolts 

In this manner, we separate the part that can be dealt with strictly in the singleparticle framework, namely h t), which is the non-interacting part, and the part that contains all the complications of electron-electron interactions. For simplicity, in the following we will discuss the case where there is only one atom per unit cell, thus eliminating the index I for the positions of atoms within the unit cell. The many-body wavefunction will have Bloch-like symmetry  

We will discuss in some detail only the case of Frenkel excitons. We will assume that we are dealing with atoms that have two valence electrons, giving rise to a simple band structure consisting of a fully occupied valence band and an empty conduction band generalization to more bands is straightforward. The many-body wavefunction will be taken as a Slater determinant in the positions ri, T2,. .. and spins si, S2. of the electrons there are N atoms in the solid, hence IN electrons in the full valence band, with one spin-up and one spin-down electron in each state, giving the many-body wavefunction 

Combination of an electron and a positive hole annihilates both defects. Maintenance of an energetic association between an electron and a positive hole, for sufficient time to permit migration of the associated pair to occur through the crystal, gives rise to a defect called an exciton. Further discussion of these defects is given in Section 1.5.1. which considers the movements of electrons in solids. 

The optical absorption of some semiconductors or insulator materials shows a series of peaks or features at photon energies close to but lower than the energy gap (the pre-edge region). These features correspond to a particular type of excitation called 

The complex of electron and hole can move with wavenumber k, and as 

The earlier formulation due to Frenkel (1931) is appropriate to molecular crystals where the overlap between the orbitals even of excited states is weak. If t i(q) is the wave function for a molecule in its ground state and ft(q) that in an excited state, then a Slater determinant Wa formed from the product 

Thus in either formulation the exciton spectrum consists of a series of bands, but the optical absorption spectrum consists of a series of lines because the selection rule 

There is a wide literature on excitons an early review is that by Knox (1963), and a book dealing exhaustively with the subject is Rashba and Sturge (1982). The aspects most relevant to the discussions in the present book are 

In nanocrystals with average radii typically below 10 nm, the band gap increases due to confinement. This is shown in Fig. 3.10 for the excitonic gap (the energy required to create an exciton) of CdS [94]. 

The review by Yoffe [106] provides a good account of the optical properties of nanocrystals in compound semiconductors (see also [89]). 

The absorption due to the formation of direct excitons associated with the T7 CB (see Fig. 3.4) have also been observed at energies above Eg in very thin silicon and germanium samples [63], and for germanium, ) of the 

The FEs produced at low temperature by illumination with photons in the vicinity or above Eg have finite lifetimes that depend on temperature (see [34] for silicon), their binding energies, and on the band structure of the semiconductor (the lifetime is larger in semiconductors with indirect gap than direct gap). During their lifetime, they can diffuse in the crystal and be trapped by impurities and defect to become bound excitons (BEs) with energies slightly different from that of the FE. 

This concept of polarons and, in particular, excitonic polarons has been used to explain observed features of the one-dimensional conducting organic materials based on 7,7,8,8-tetracyano-p-quinodimethane, TCNQ (85). It indicates that a way to reduce the Coulomb repulsion between electrons in the chain is to surround each chain by a highly polarizable medium. However, a limit may be reached beyond which, if the surrounding medium were made more polari zable, the effects due to band narrowing would outweigh the benefits of reduced Coulomb repulsion (85). 

Absorption coefficients for Si and GaAs. The band-gap edges are located by vertical lines. (Absorption coefficients were computed from Palik, E.D., Ed., Handbook of Optical Constants of Solids, Vols. 1-3, Academic Press, 1997.) 

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Figure Al.3.25. Schematic illustration of exciton binding energies in an insulator or semiconductor.

Even in semiconductors, where it might appear that the exciton binding energies would be of interest only for low temperaPire regimes, excitonic effects can strongly alter tlie line shape of excitations away from the band gap. 

The size of the exciton is approximately 50 A in a material like silicon, whereas for an insulator the size would be much smaller for example, using our numbers above for silicon dioxide, one would obtain a radius of only 3 A or less. For excitons of this size, it becomes problematic to incorporate a static dielectric constant based on macroscopic crystalline values. 

The reflectivity of LiF is illustrated in figure Al.3.26. The first large peak corresponds to an excitonic transition. 

The SHG/SFG technique is not restricted to interface spectroscopy of the delocalized electronic states of solids. It is also a powerful tool for spectroscopy of electronic transitions in molecules. Figure Bl.5.13 presents such an example for a monolayer of the R-enantiomer of the molecule 2,2 -dihydroxyl-l,l -binaphthyl, (R)-BN, at the air/water interface [ ]. The spectra reveal two-photon resonance features near wavelengths of 332 and 340 mu that are assigned to the two lowest exciton-split transitions in the naphtli-2-ol... 

Sonnenschein R, Syassen K and Otto A 1981 Effect of pressure on the first singlet exciton in crystalline anthracene J. Chem. Phys. 74 4315... 

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... 

Kopelman R, Tan Wand Birnbaum D 1994 Subwavelength spectroscopy, exciton supertips and mesoscopic light-matter interactions J. Lumin. 58 380-7... 

Bach FI, Renn A, Zumofen G and Wild U P 1999 Exciton dynamics probed by single molecule spectroscopy Phys. Rev. Lett. 82 2195-8... 

Higgins D A and Barbara P F 1995 Excitonic transitions in J-aggregates probed by near-field scanning optical microscopy J. Chem. Phys. 99 3-7... 

Figure C2.17.10. Optical absorjDtion spectra of nanocrystalline CdSe. The spectra of several different samples in the visible and near-UV are measured at low temperature, to minimize the effects of line broadening from lattice vibrations. In these samples, grown as described in [84], the lowest exciton state shifts dramatically to higher energy with decreasing particle size. Higher-lying exciton states are also visible in several of these spectra. For reference, the band gap of bulk CdSe is 1.85 eV.

An explanation for these size-dependent optical properties, tenned quantum confinement , was first outlined by Bms and co-workers in the early 1980s, [156, 158, 159, 160 and 161] and has fonned the basis for nearly all subsequent discussions of these systems. Though recent work has modified and elaborated on this simple model, its basic predictions are surjDrisingly accurate. The energy of the lowest-lying exciton state is given by the following simple fonnula ... 

Here, E and s are the band gap energy and the dielectric constant of the bulk semiconductor, and p is the reduced 0 mass of the exciton system, 1/p = + 1/fffi,. The second tenn, proportional to /R, arises from a simple... 

Figure C2.17.11. Exciton energy as a function of particle size. The Bms fonnula is used to calculate the energy shift of the exciton state as a function of nanocrystal radius, for several different direct-gap semiconductors. These estimates demonstrate the size below which quantum confinement effects become significant.

Figure C2.17.12. Exciton energy shift witli particle size. The lowest exciton energy is measured by optical absorjDtion for a number of different CdSe nanocrystal samples, and plotted against tire mean nanocrystal radius. The mean particle radii have been detennined using eitlier small-angle x-ray scattering (open circles) or TEM (squares). The solid curve is tire predicted exciton energy from tire Bms fonnula.

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. 

Leung K, Pokrant S and Whaley K B 1998 Exciton fine structure in CdSe nanoclusters Phys. Rev. B 57 12 291... 

Let us now consider the case where tliere is more tlian one exciton in tlie given molecular ensemble. The presence of two or more excess excitons not only creates two or more holes in tlie ground state (see case (a) above) but it also opens up tlie possibility of two excitons being found on neighbouring molecules. Then tlie following two-stage process can take place [26] ... 

In tlie first stage, where at first we have two excitons S, excitation jumps from one of tlie excited molecules to... 

S-S annihilation phenomena can be considered as a powerful tool for investigating tire exciton dynamics in molecular complexes [26]. However, in systems where tliat is not tire objective it can be a complication one would prefer to avoid. To tliis end, a measure of suitably conservative excitation conditions is to have tire parameter a< )T < 0.01. Here x is tire effective rate of intrinsic energy dissipation in tire ensemble if tire excitation is by CW light, and T = IS tire... 

Recent tlieoretical [35, 36 and 37] and experimental [38] research has revealed anomalous behaviour of tire dimer anisotropy under certain excitation conditions. If tire dimer is excited by broadband light tliat covers botli excitonic transitions, or by a relatively narrow band properly positioned between tire maxima of tire excitonic transitions, tire... 

With tlie development of femtosecond laser teclmology it has become possible to observe in resonance energy transfer some apparent manifestations of tire coupling between nuclear and electronic motions. For example in photosyntlietic preparations such as light-harvesting antennae and reaction centres [32, 46, 47 and 49] such observations are believed to result eitlier from oscillations between tire coupled excitonic levels of dimers (generally multimers), or tire nuclear motions of tire cliromophores. This is a subject tliat is still very much open to debate, and for extensive discussion we refer tire reader for example to [46, 47, 50, 51 and 55]. A simplified view of tire subject can nonetlieless be obtained from tire following semiclassical picture. 

Valkunas L, Trinkunas G and Liuolia V 1998 Exciton annihilation in molecular aggregates Resonance Energy Transfer ed D L Andrews and A A Demidov (New York Wiley) pp 244-307... 

Kolubayev T, Geacintov N E, Paillotin G and Breton J 1985 Domain sizes in chloroplasts and chlorophyll-protein complexes probed by fluorescence yield quenching induced by singlet-triplet exciton annihilation Biochimica Biophys. Acta 808 66-76... 

Tinoco I 1963 The exciton contribution to the optical rotation of polymers Radiat. Res. 20 133-9... 

In order to demonstrate the NDCPA a model of a system of excitons strongly coupled to phonons in a crystal with one molecule per unit cell is chosen. This model is called here the molecular crystal model. The Hamiltonian of... 

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