Indirect transition

Trigonal selenium is variously called metallic gray or black selenium and occurs in lustrous hexagonal crystals, which melt at 220.5 °C. Its structure, which has no sulfur analogue, consists of infinite, unbranched helical chains. Its density, 4.82 g cm , is the highest of any form of the element. Trigonal selenium is a semiconductor (intrinsic p-type with a rather indirect transition at about 1.85 eV [5]), and its electronic and photoelectric properties are the basis for many industrial uses of this element. 

Optical band gap energies (Eg) for WOx-ZrOa samples calcined at 1073 K were obtained from UV-vis spectra using procedures based on direct and indirect transitions between valence and conduction bands [26]. Direct band gap energies (Egdecreased monotonically from 4.15 to 3.75 eV as the W loading increased from 3.05 to 15.0 W-atomsnm (Table 2). 

In materials with a band structure such as that sketched in Figure 4.8(b), the bottom point in the conduction band has a quite different wave vector from that of the top point in the valence band. These are called indirect-gap materials. Transitions at the gap photon energy are not allowed by the rule given in Equation (4.29), but they are still possible with the participation of lattice phonons. These transitions are called indirect transitions. The momentum conservation rule for indirect transitions can be written as... 

Indirect transitions are much weaker thau direct trausitious, because the latter do uot require the participation of photons. However, many indirect-gap materials play an important role in technological applications, as is the case of silicon (band structure diagram iu Figure 4.7(a)) or germanium (baud structure diagram shown later, in Figure 4.11). Hereafter, we will deal with the spectral shape expected for both direct and indirect transitions. 

Because of the involvement of phonons in indirect transitions, one expects that the absorption spectrum of indirect-gap materials must be substantially influenced by temperature changes. In fact, the absorption coefficient must be also proportional to the probabihty of photon-phonon interactions. This probabihty is a function of the number of phonons present, t]b, which is given by the Bose-Einstein statistics ... 

In Ref. 54, XRD showed the deposit to be hexagonal CuSe. Analysis of the absorption spectrum gave a direct bandgap of 2.02 eV. As commonly seen for these compounds, there was still strong absorption at lower energies (e.g., at 1.9 eV, the absorption coefficient was >7 X 10" cm ), possibly due to an indirect transition but likely due, at least in part, to free-carrier absorption. From Hall measurements, the doping (acceptor) density was ca. 10 cm (heavily degenerate) and the mobility ca. 1 cm V sec The dependence of film thickness and deposition rate on the deposition parameters has been studied in a separate paper [62]. 

Fig. 32. Diagram of interband electron transitions under photoemission 1—direct transition, 2—indirect transition.

The photoconductivity and absorption spectra of the multilayer polydiacetylene are shown in Fig. 22 [150]. The continuous and dotted line relate to the blue and red polymer forms respectively. Interpretation is given in terms of a valence to conduction band transition which is buried under the vibronic sidebands of the dominant exciton transition. The associated absorption coefficient follows a law which indicates either an indirect transition or a direct transition between non-parabolic bands. The gap energies are 2.5 eV and 2.6 eV for the two different forms. The transition is three dimensional indicating finite valence and conduction band dispersion in the direction perpendicular to the polymer chain. 

Karavaev, G. F. (1965) Selection rales for indirect transitions in crystals. Sov. Phys. Solid-State 6, 2943-8. 

Case A Indirect Transition to Isolated Resonance Here the photodissociation occurs by excitation to a single resonance, followed by a transition from the resonance to the continuum. In this case the sum over s reduces to a single term, and the direct optical transitions to the continuum are suppressed. That is,... 

Case D Sum of Direct and Indirect Transition to Isolated Resonance ... 

For an indirect transition, the transition rate is expressed as follows [11] ... 

It indicates that the absorption coefficient for an indirect transition to energy E from an initial energy (see Eq. (1)) is proportional to the product of the initial density of states and the final density of states. When S3 absorbs a light having photon energy over the band gap, electrons in N 2p in the valence band are excited to Hf 5d related to Hf-N bonds in the conduction band. It is speculated that the increase of the absorption coefficient is low because the pDOS of Hf 5d and N 2p related to Hf-N bonds is small. When a photon with energy over 3.8 eV is absorbed, it is expected that the absorption coefficient increases abruptly because the pDOS of Hf 5d and N 2p related to Hf-N bonds is large. 

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 ... 

Depending upon the relationship between the momentum in the initial and final states (which, in turn, depend on the profile of the parabolic energy valley ) direct or indirect transitions can occur, as shown in Figure 2.4, and this affects all the three terms Pn, vk and Uf. It must be noted that in transitions between indirect valleys (Figure 2.4B) momentum is conserved via interaction with a phonon (i.e. a quantum lattice vibration), which can be either emitted or adsorbed. Some additional detail on such transitions will be given in the section deaUng with semiconductor oxides. 

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