Band Structure Considerations

How both the density and mobility of charge carriers in metals and band semiconductors (i.e. those in which electrons are not localized by disorder or correlation) are influenced by particular features of the electronic structure, namely band dispersion and band Ailing, will now be examined. Taking mobUity first, this book will briefly revisit the topic of band dispersion. Charge carriers in narrow bands have a lower mobility because they 

Extrinsic semiconductors ate those in which the carrier concentration, either holes or electrons, are controlled by intentionally added impurities called dopants. The dopants are termed shallow impurities because their energy levels lie within the band gap close to one or other of the bands. Because of thermal excitation, -type dopants (donors) are able to donate electrons to the conduction band and p-type dopants (acceptors) can accept electrons from the valence band, the result of which is equivalent to the introduction of holes in the valence band. Band gap widening/narrowingmay occur if the doping changes the band dispersion. At low temperamres, a special type of electrical transport known as impurity conduction proceeds. This topic is discussed in Section 7.3. 

Although scattering processes in both semiconductors and metals increase with rising temperamre, thereby decreasing the mobility of the carriers, the scattering is more than offset in a semiconductor by an increase in the charge carrier concentration. Thus, the electrical resistivity of a semiconductor decreases with increasing temperamre, dp/dT 0. In a metal, the resistivity has the opposite temperamre dependency dp/dT 0. 

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The mixed valency situation may be equally well described in terms of bands. The band to be considered in the case of bismuth is the Bi6s-02p band. The band in the case of copper is the Cu3d y2 -02p band. Both of these are d bands. For the lowest oxidation state (Cu1 or Bira), these bands are filled for the highest oxidation state (Cu111 or Biv), these bands are empty. According to simple band structure considerations, we would expect metallic properties for any partial filling of these o bands in bismuth and copper oxides. While metallic properties are indeed observed for most of the intermediate... 

Al) aimed to reduce the Co content, and stabilize the structure to reversible lithium extraction. Materials with high nickel content LiNii j,M M j,02 are appealing, but suffer from oxidation reactions with the electrolyte at elevated temperature/voltage as does LiNi02. Band structure considerations indicate that extraction of oxygen from the lattice competes with removal of Li+ at high potential, in the order Ni > Co > Mn. ... 

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

I be second important practical consideration when calculating the band structure of a malericil is that, in principle, the calculation needs to be performed for all k vectors in the Brillouin zone. This would seem to suggest that for a macroscopic solid an infinite number of ectors k would be needed to generate the band structure. However, in practice a discrete saaipling over the BriUouin zone is used. This is possible because the wavefunctions at points... 

The main difference between the two models lies in the fitted scattering rate T (Table 2) which is considerably smaller, one order of magnitude, for the BM then for the MG model. Moreover, we also notice that (Op(a ) > (Oy,(aj.) and that cOp is slightly larger in BM compared to MG. According to the band structure... 

Valence band spectra provide information about the electronic and chemical structure of the system, since many of the valence electrons participate directly in chemical bonding. One way to evaluate experimental UPS spectra is by using a fingerprint method, i.e., a comparison with known standards. Another important approach is to utilize comparison with the results of appropriate model quantum-chemical calculations 4. The combination with quantum-chcmica) calculations allow for an assignment of the different features in the electronic structure in terms of atomic or molecular orbitals or in terms of band structure. The experimental valence band spectra in some of the examples included in this chapter arc inteqneted with the help of quantum-chemical calculations. A brief outline and some basic considerations on theoretical approaches are outlined in the next section. 

With respect to the physical properties mentioned, band-structure calculations have attracted considerable interest, e.g., for SbSBr, SbSI, and SbSel (234). For the compounds having reference 22i in column 4 of Table XXIX, a temperature-independent diamagnetism has been found, with values of about 10 cm" g between 77 and 340 K. A small temperature-dependence is exhibited by BiTel, a narrow-gap semiconductor (41). The anisotropy of the magnetic susceptibility has been studied for SbSI, BiSel, and BiTel (41, 420). 

Tl2Ba2Cu06 (23) yielding a means of oxidizing the Cu02 sheets. Also, recent band structure calculations (discussed below) show that there is considerable overlap of the T16s-block bands with the Cu 3d x2-y2 bands providing a mechanism to transfer electrons from the Cu02 sheets to the Tl-O layers. 

The assumption that the carriers are small or intermediate polarons in no way militates against discussions of ths band structure of the ground state (see e.g. Camphausen et al. 1972, Cullen and Callen 1971,1973). The absence of Jahn-Teller distortion (Goodenough 1971) also, in our view, indicates not the absence of a polaron mass-enhancement but rather a value.of V0jB not too far from the critical value. These conclusions seem to be in agreement with the considerations of Sokoloff (1972), who used a description in terms of a degenerate band of small polarons. Samara (1968) showed that pressure lowers the temperature of the Verwey transition. If this depended only on e2/ ca then the opposite should be the case. But pressure will increase B, and push the substance nearer to the critical value for the metal-insulator transition. 

Haaland also measured the infrared spectra of benzene adsorbed on Pt/Al203 that had been regenerated after previous benzene/cyclohexane adsorptions (247) the surface was thought to retain structured carbonaceous deposits. In this case, the broad yCH feature was centered at ca. 3030 cm 1 (with components at 3042, 3031, 3024, and 3014 cm 1) rather than 3040 cm 1 for the species on the freshly prepared catalyst, and a weaker companion band occurred at 2947 cm-1. The benzene absorption bands at wavenumbers <1500 cm 1 were little changed in position but become more prominent in room-temperature spectra in which the 2947-cm 1 feature was weakened. Spectra measured over the range 300-650 K showed that the 2947-cm 1 feature disappeared at 435 K, whereas the vCH aromatic bands retained considerable intensity at temperatures up to 560 K. 

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