Direct interband transition

In Table 2, the calculated values of energy maxima E, and band areas Si of transverse transitions were compared for the three variants of calculations of optical function sets. These variants differ comparatively little in energy El. The principal inconsistencies are observed in the values of Si. This is caused by the differences in the experimental R(E) spectra, i.e. by using of ZnO samples of different quality and by different registration techniques for polarized reflectivity spectra. The determined components of S2 and -Ims spectra are caused by direct interband transitions or metastable excitons, except for the most long-wavelength of them, which are associated with free excitons. The theoretical band calculations of ZnO ° strongly differ in the bands dispersion and positions. This makes it difficult to propose a... 

Theoretical studies were performed for Using a simple microscopic theory 2 is correlated to direct interband transitions. C2((o) can be calculated using a simple approach from the joint density of states J (to), see below. 2(0)) is then ... 

In the given paper we present a quantum-mechanical theory of HHG in CNs utilizing a single electron approximation in the tight-binding model. Our approach is based on the previous study HHG in CNs presented in Refs. [1,2] and allows us to incorporate into consideration direct interband transitions. We have calculated an axial current density, spectrum of which is responsible for HHG in CNs. 

The optical properties have been calculated from reflectance spectra of Si(l 11)/NC CrSi2/Si system with a monocrystalline structure. Calculations of the absorption coefficients permit to determine the energies of direct interband transitions in CrSi2 nanocrystallites. Some dispersion of these energies (1.8-2.9 eV) testifies about contributions of NC s with different sizes. 

Pristine poly(acetylene) in either cis- or trans-form exhibits optical spectra typical for a semiconductor. The absorption peak of the cis-polymer is structured with maxima at 2.1, 2.3 and 2.4 eV, the absorption coefficient is about 4 10 cm" The structureless and very broad absorption of the trans-polymer peaks at 1.9 eV with an absorption coefficient of 3 10 cm . The large halfwidth of the peak is generally explained as being due to the rather broad chain length distribution in addition, the small crystallite size and crystal defects may contribute to the broadening. A discussion is still ungoing whether the peak is linked to a direct interband transition or whether it is of an excitonic character.2-4, 7,32... 

The classical theory of the dielectric response in solids is frequently described by the Drude and Lorentz models. The Drude model is applicable to free-eiectron metals its quantum-meehanical analog includes intraband transitions, where intraband transitions are taken to mean all transitions not involving a reciprocal lattice vector. The Lorentz model is applicable to insulators its quantum-mechanieal analog ineludes all direct interband transitions, i.e., all transitions for whieh the final state of an electron lies in a different band but with no change in the k vector in the reduced-zone scheme. In the following discussion, both models will be surveyed and evaluated for real metals. 

The Lorentz and Drude models can be explained rigorously in relation to electronic band structure. Indeed, both models have quantum-mechanical analogs intraband transitions for the Drude model and direct interband transitions for the Lorentz model. To see the role of both models in describing real metals, consider the schematic band diagram as shown in Figure 149. TWo typical transitions are illustrated in Figure 149. The first of these, called an intraband transition, corresponds to the optical excitation of an electron from below the Fermi level (Ep) to another state above the Ef within the same band. There is no threshold energr for such transitions, and they can occur only in metals. 

The results obtained may be interpreted starting from the analysis of polarisation effects and the selection rules, which can be established using the model of direct interband transitions based on LAPW calculations of the TiC electronic structure (Fig. 8.2). As has been found by Callenas (1985), peak D is related to the transition between between the two Ai symmetry states Aj( p — 2.9 eV) and Ai( p + 18.3 eV). Peak C which is... 

We next consider the effect of direct interband transitions, that is, transitions for which k = k and n n. To ensure direct transitions we introduce a factor (27r) 5(k — kO in the double summation over wave-vectors in the general expression for the dielectric function, Eq. (5.47), from which we obtain... 

Reflectance spectrum of Si. Peaks at 3.4, 4.5, and 5.4 eV are direct interband transitions. (Reflectivity was computed from Palik, E.D., Ed., Handbook of Optical Constants of Solids, Vols. 1-3, Academic Press, 1997.)... 

This important selection rnle indicates that interband transitions mnst preserve the wave vector. Transitions that preserve the wave vector (snch as those marked by vertical arrows in Figure 4.8(a)) are called direct transitions, and they are easily observed in materials where the top point in the valence band has the same wave vector as the bottom point in the conduction band. These materials are called direct-gap materials. 

Figure 4.8 Interband transitions in solids with band-gap energy Eg-, (a) A direct band gap. Two direct transitions are indicated by arrows, (b) An indirect band gap. Two indirect band-gap transitions are indicated by arrows. The transitions at photon energies lower than Eg require absorption of phonons. The transitions at photon energies higher than Eg involve emission of phonons.

The minus (plus) sign corresponds to E°7+(E°7.)- The two splitting energies are directly measurable quantities from the difference between the interband transition energies in the optical measurement or theoretical calculation. Ac, and A, are obtainable from such data via fitting to the above theoretical equations. 

Of particular interest are the optical spectra. Chclikow.sky and Schluter calculated the Joint density of states for direct transitions (which would be proportional to C2 were the dipole matrix elements all equal). sec Section 4-A -with the result shown at the bottom of Fig. 11-12. It bears little resemblance to the experimental Cj curve (uppermost in the figure), for a number of reasons. Tlie prominent peak at 10.4 eV appears to be an cxciton peak (See Section 6-H), as had been stiggested earlier by Platzoder (1968) on the basis of observed temperature dependence. Pantelidcs and Harrison took this peak to result from interband transitions, since it lay at an enci gy above the photoconductivity threshold of 9 eV (DiStephano and Eastman, 1971b) that would rule out the possibility that the peak represents a simple exciton, but not that it represents an excitonlikc... 

Regarding the fundamental interband transition and the corresponding photogeneration of electron-hole pairs, the interband transitions have to be divided into direct and indirect transitions. The meaning of these terms is as follows ... 

The large joint density of states associated with the direct it- to ir -(interband) transition results in a very strong cross-section for stimulated emission. 

The SPR is then also called Mie resonance. For simple metals, the SPR absorption band has a Lorentzian shape peaked at oi p, the width of which is directly proportional to the collision constant E introduced in the Drude description of the metal dielectiic constant (Eq. 2). Of course, for noble metals the absorption due to interband transitions has to be taken into account in order to obtain the complete spectrum. 

The optical properties of (CH) are of interest, since these directly give information about the band gap and/or the levels in the midgap, which is considered to be closely related with the electric transport property. Currently available experimental data on the photoabsorption of (CH), polymers are listed in Table IV. It seems to be reasonable to regard these absorption onsets as the it - tt interband transition energy from the HO to the LU band, that is, the band gap between the valence and the conduction bands in the sense of the one-electron approximation based on the one-dimensional Peierls transition mentioned in the Section II,A. 

Most wurtzite-type crystals are direct band-gap materials (2fP-SiC is an exception) and interband transitions can take place between these three Fils and the T7 CB minimum. These materials are anisotropic and this anisotropy reflects on the selection rules for the optical transitions and on the effective masses. The Tg (A) —> T7 (CB) transitions are only allowed for ETc while the two T7 (B. C) —> T7 (CB) transitions are allowed for both polarizations. However, the relative values of the transition matrix elements for the T7 (B, C) —> T7 (CB) transitions can vary with the material. For instance, in w-GaN, the T7 (B) —> T7 (CB) transition is predominantly allowed for ETc while the T7 (C) — I 7 (CB) transition is predominantly allowed for E//c [22]. Table 3.7 gives band structure parameters of representative materials with the wurtzite structure. 

There are several perspective directions of the development of the presented theory. First, indirect interband transitions of 7i-electrons should be included in the consideration they give rise to the transverse current in CNs. Second, the HHG theory should be generalized for the case of multi-wall CNs or CN-composites. Third, effects of chirality should be taken into account. 

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