Surface excitons dispersion

The modification of the surface excitonic dispersion is inexplicable in the absence of reconstruction. 

Figure 2.18. Profile of isoenergetic surfaces of the excitonic dispersion in the vicinity of the bottom of the band in the model (2.139) of an orthorhombic crystal. We note the lengthening of the surfaces along c . Also, the wave vectors tend to orient perpendicular to d (or the b axis) in the vicinity of the point K = 0.

For the b surface exciton, with its dipole contained in the 2D lattice layer, the nonanalytic part of the dispersion has the value (d2/2 0S0)K cos2(K,d). For... 

The plane waves of a perfect 2D lattice diagonalize the electromagnetic interactions, giving rise to the excitonic dispersion through the Brillouin zone, and to the surface-exciton-polariton phenomenon around the zone center.148,126 The corresponding hamiltonian may be written as... 

One of the main theoretical problems is to determine the dependence of the energy of a surface exciton on its wavevector or, in other words, to obtain the dispersion law for surface excitons. Then the next problems arise in consideration of the interactions of these excitons with light, phonons and with surface defects. 

As a matter of fact, as can be seen from eqn (12.17), the limit k —> oo is consistent with the sum t (u>) + e2(w) approaching zero, which in the particular case of the boundary with vacuum (ei = 1, e2 = e) agrees with eqn (12.12). The result of this limiting transition confirms once more the remark made above, viz. that surface polaritons for large values of k transfrom into Coulomb surface excitons. The dispersion law for Coulomb surface excitons at a sharp boundary and without taking spatial dispersion into account has the form... 

As stressed earlier, the transition from surface polaritons to Coulomb surface excitons corresponds to the limiting transition c —> oo. For p-polarized waves it yields the dispersion relation... 

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. 

Although some optical techniques, such as soft X-ray absorption and optical reflectance measurements, provide comparative information about solids with higher energy resolution, EELS enjoys several unique advantages over optical spectroscopies. First of all, unlike optical reflectance measurements which are sensitive to the surface condition of the sample, the transmitted EELS represents the bulk properties of the material. Secondly, EELS spectra can be measured with q along specific controllable directions and thus, can be used to study the dispersion of plasmons, excitons, and other excitations [8.1-8.5]. Such experiments offer both dynamics as well as symmetry information about the electronic excitations in solids. In addition, the capability to probe the electronic structure at finite momentum-transfer also allows one to investigate the excited monopole or quadrupole transitions, which cannot be directly observed by conventional optical techniques limited by the dipole selection rule. 

One major difference between a semiconductor nanocluster and molecule is the size dependence of a. As discussed in Section II.B and illustrated in Figure 5, there is a rapid rise in the absorption cross section of the first excited state as the cluster size is reduced below the exciton size. The resonant third-order nonlinearity of a nanocluster is therefore predicted to increase with decreasing cluster size. In reality it is limited by the cluster size dispersion and the presence of surface defects [15]. To maximize the resonant nonlinearity, one needs (1) a sharp exciton absorption band (which means smaller and monodisperse clusters), and (2) semiconductor clusters with larger absorption coefficients, such as GaAs or PbS. 

Nano-composite materials with fine semiconductor particles dispersed in the matrix have attracted considerable interest because the properties of the particles are much different from their bulks when the diameters are l s than the Bohr exciton radius. Such particles, which are generally named as nano-particles, are characterized by non-stoichiometric surface structure and quantum size effect 2). These properties would lead to new phenomena, new theoretical insights, and new materials and devices. 

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