Conduction indirect transitions

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... 

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. 

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. 

The origin of the intense 248-nm emission without band A has not been elucidated yet. A speculation is that the B impurity band is extended deep inside the Brillouin zone, and a direct transition from the conduction band minimum to the impurity band is involved in addition to the ordinal indirect transition... 

Figure 3. The nonrelativistic band structure for AgBr. The lowest energy indirect transition promotes an electron from the highest point in the valence band, L 3, to the minimum in the conduction band, T,. This energy difference is the bandgap energy, Et. The lowest energy direct transition is from r,5 to T,. After [52] and [54],...

Figure 14.9 (a) In a direct transition between the valence band and the conduction band, the wave vector, k, of the excited electron does not change (b) in an indirect transition, the wave vector, k, changes, making the transition much less probable than a direct transition... 

Electronic transitions between the valence and conduction bands in the crystal start at the absorption edge which corresponds to the minimum energy difference Eg between the lowest minimum of the conduction band and the highest maximum of the valence band. If these extrema lie at the same point of the k-space, the transitions are called direct. If this is not the case, the transitions are possible only when phonon-assisted and are called indirect. The rule governing these transitions is the conservation of quasimomentum during the transitions, either of the electron alone in direct transitions, or the sum of the electron and phonon quasimomenta in indirect transitions. 

The luminescence mainly originates from the inter-band transitions, which are divided into direct and indirect transitions according to the transition modes. If the electrons jump at the same point between the VBM (valence band maximum) and the CBM (conduction band minimum), this transition is direct. In contrast, there is indirect transition. The semiconductors silicon (Si) and gallium arsenide (GaAs) are typical examples as shown in Fig. 6.6. They have an indirect and direct band gap with the values 1.95 and 0.17 eV, respectively. When the crystal size becomes smaller, e.g., forming quantum dots, the Si becomes a better self-activated luminescence material. 

Crystal defects, surfaces, and some impurities create deep levels in the bandgap. These levels are separated from the conduction and valence bands by more than approximately 5 ksT. Deep levels act as neither good acceptor nor good donor levels because they are not usually ionized at room temperature. Deep levels provide an alternative mechanism for the recombination of holes and electrons and thus affect radiative efficiency. Transitions where phonons are not involved are called direct transitions indirect transitions involve phonons and are less radiatively efficient. 

The general theoretical expression for the intensities of the excitonic lines in the optical spectra, corresponding to TO phonon assisted indirect transitions between the stress-split Tg y valence and Al,c conduction bands via T] 5 conduction and A5 y valence band intermediate states are listed in Table IV. 

There are two main regions of absorption, corresponding to so-called direct and indirect transitions. The weaker absorption at longer wavelengths represents the indirect, forbidden transitions, which gain intensity via lattice vibrations (phonons). Absorption of a photon promotes an electron from the valence band to the conduction band, shown by the blue arrow in Fig. 11.8. 

Figure 5.3. Illustration of optical transitions. Left interband transitions in a semiconductor, between valence and conduction states 1 is a direct transition at the minimal direct gap, 2 is another direct transition at a larger energy, and 3 is an indirect transition across the minimal gap Cgap. Right intraband transitions in a metal across the Fermi level ep...

Figure 38. Model of the band diagram of the passive oxide on iron for photo excitation proeess. The flat band is about 0.35 V vs. RHE and the band gap between the valenee and conduction band edges is about 2.6 eV. The direct transition may take plaee over this band gap energy. The indirect transition may take place via excitation from the valence band to the mid gap level i.e. the ionized donor sites with the exeitation energy at about 2 eV.

Schematic of direct and indirect bandgap transition. The vertical transitions are allowed direct transitions. The indirect transition from the fc = 0 to the minimum of the conduction band requires the photon to combine with a phonon in order to conserve momentum.

In direct bandgap materials, electrons may go directly from the conduction band to the valence band with a photon emission, which makes them suitable as LEDs or solid-state lasers. Indirect transitions from the conduction to the valence band must be accompanied by the emission of a phonon and are nonradiative transitions, which results in heating the material. 

For an indirect transition (where the momentum of the electron in the conduction band is not the same as the hole in the valence band and a phonon as well as a photon is required this lowers the probability of the transition and therefore the value of a) ... 

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