Electronic band structure for

Figure 7.7a shows the extended-zone electronic band structure for a one-dimensional crystal - an atom chain with a real-space unit cell parameter a and reciprocal lattice vector Tr/n - containing a half-filled (metallic) band. In this diagram, both values of the wave vector, +k, are shown. The wave vector is the reciprocal unit cell dimension. The Fermi surface is a pair of points in the first BZ (Fig. 7.7c). When areas on the Fermi surface can be made to coincide by mere translation of a wave vector, q, the Fermi surface is said to be nested. The instability of the material towards the Peierls distortion is due to this nesting. In one dimension, nesting is complete and a one-dimensional metal is converted to an insulator because of a Peierls distortion. This is shown in Figure 7.7b, where the real-space unit cell parameter of the distorted lattice is 2a and a band gap opens at values of the wave vector equal to half the original values, 7r/2a.

The photocatalytic properties of inorganic semiconductors strongly depend on the electronic band structure. For photochemical water reduction to occur, the flat-band potential of the semiconductor (for highly doped semiconductors, this equals the bottom of the conductance band) must exceed the proton reduction potential of 0.0 V vs NHE at pH = 0 (-0.41 V at pH = 7 Figure lA). To facilitate water oxidation, the potential of the valence band edge must exceed the oxidation potential of water of +1.23 V vs NHE at pH... 

Fig. 7.29 Electronic band structures for Mg2NiH4 calculated with LDA in (a) the low-temperature monoclinic phase and (b) the high-temperature cubic phase with undistorted tetrahedral NiH4 complexes. (From Myers et al. (2002), Ref [192].)...

Understanding or predicting the nature of the electronic band structure for a periodic material requires that we identify a unit cell and accompanying basis set in real space and then move this basis set along translation vectors a, possibly with accompanying... 

Of the three parameters that determine the conductivity, the effective carrier mass, m, is most closely related to the details of the electronic band structure. For a carrier at a given point in a band structure, the effective mass is related to the curvature of ih E k) function by the equation ... 

FIGURE 3.55 Electronic band structure for metals, semiconductors, and insulators. 

Fig. 7.3 (Color online) Electronic band structure for a-Po. Here we show our results with SO coupling (solid line) and without SO coupling (dashed line)...

Describe the four possible electron band structures for solid materials. 

Fig. 26.11. Proposed electronic band structure for cerium hydride (Libowitz et al., 1972).

Surface electronic band structures for the GaN and InN (0001) surfaces are shown in Figure 13.41a,b, respectively. The band structures correspond to (2 x 2) Ga adatom structures where the Ga(In) adatom binds to three surface Ga(In) atoms and one surface Ga (surface In) remains threefold-coordinated on the (0001) surface. On GaN, the adatom structure leads to a semiconducting surface band structure with two separate occupied and empty electronic bands with very low dispersion. The microscopic origin of these bands is accessible from the calculation as well. The upper empty state stems from dangling bonds of the Ga adatoms and from the dangling bonds of the threefold-coordinated Ga surface atom. The lower occupied state arises from the bonding between the Ga adatoms and the three Ga surface atoms below. In the case of the InN surface, the surface electronic... 

As an example of a nanotube representative of the diameters experimentally found in abundance, we have calculated the electronic structure of the [9,2] nanotube, which has a diameter of 0.8 nm. Figure 8 depicts the valance band structure for the [9,2] nanotube. This band structure was calculated using an unoptimized nanotube structure generated from a conformal mapping of the graphite sheet with a 0.144 nm bond distance. We used 72 evenly-spaced points in the one-... 

Seebeck used antimony and copper wires and found the current to be affected by the measuring instrument (ammeter). But, he also found that the voltage generated (EMF) was directly proportional to the difference in temperature of the two junctions. Peltier, in 1834, then demonstrated that if a current was induced in the circuit of 7.1.3., it generated heat at the junctions. In other words, the SEEBECK EFFECT was found to be reversible. Further work led to the development of the thermocouple, which today remains the primary method for measurement of temperature. Nowadays, we know that the SEEBECK EFFECT arises because of a difference in the electronic band structure of the two metals at the junction. This is illustrated as follows ... 

A list of recent solid-state calculations is given in Refs. [43-45]. We mention only a few of the most recent results discussing relativistic effects. Christensen and Kolar revealed very large relativistic effects in electronic band structure calculations for CsAu... 

Band structure for a chain of H atoms. Left, with equidistant atoms right, after PEIERLS distortion to H2 molecules. The lines in the rectangles symbolize energy states occupied by electrons... 

The structure of MnP is a distorted variant of the NiAs type the metal atoms also have close contacts with each other in zigzag lines parallel to the a-b plane, which amounts to a total of four close metal atoms (Fig. 17.5). Simultaneously, the P atoms have moved up to a zigzag line this can be interpreted as a (P-) chain in the same manner as in Zintl phases. In NiP the distortion is different, allowing for the presence of P2 pairs (P ). These distortions are to be taken as Peierls distortions. Calculations of the electronic band structures can be summarized in short 9-10 valence electrons per metal atom favor the NiAs structure, 11-14 the MnP structure, and more than 14 the NiP structure (phosphorus contributes 5 valence electrons per metal atom) this is valid for phosphides. Arsenides and especially antimonides prefer the NiAs structure also for the larger electron counts. 

For the conduction electrons, it is reasonable to consider that the inner-shell electrons are all localized on individual nuclei, in wave functions very much like those they occupy in the free atoms. The potential V should then include the potential due to the positively charged ions, each consisting of a nucleus plus filled inner shells of electrons, and the self-consistent potential (coulomb plus exchange) of the conduction electrons. However, the potential of an ion core must include the effect of exchange or antisymmetry with the inner-shell or core electrons, which means that the conduction-band wave functions must be orthogonal to the core-electron wave functions. This is the basis of the orthogonalized-plane-wave method, which has been successfully used to calculate band structures for many metals.41... 

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