Insulating electronic structures

In this chapter the results of detailed research on the realistic electronic structure of single-walled CNT (SWCNT) are summarised with explicit consideration of carbon-carbon bond-alternation patterns accompanied by the metal-insulator transition inherent in low-dimensional materials including CNT. Moreover, recent selective topics of electronic structures of CNT are also described. Throughout this chapter the terminology "CNT stands for SWCNT unless specially noted. 

It is well known that metallic electronic structure is not generally realised in low-dimensional materials on account of metal-insulator transition (or Peierls transition [14]). This transition is formally required by energetical stabilisation and often accompanied with the bond alternation, an example of which is illustrated in Fig. 4 for metallic polyacetylene [15]. This kind of metal-insulator transition should also be checked for CNT satisfying 2a + b = 3N, since CNT is considered to belong to also low-dimensional materials. Representative bond-alternation patterns are shown in Fig. 5. Expression of band structures of any isodistant tubes (a, b) is equal to those in Eq.(2). Those for bond-alternation patterned tube a, b) are given by. 

Electronic structures of SWCNT have been reviewed. It has been shown that armchair-structural tubes (a, a) could probably remain metallic after energetical stabilisation in connection with the metal-insulator transition but that zigzag (3a, 0) and helical-structural tubes (a, b) would change into semiconductive even if the condition 2a + b = 3N s satisfied. There would not be so much difference in the electronic structures between MWCNT and SWCNT and these can be regarded electronically similar at least in the zeroth order approximation. Doping to CNT with either Lewis acid or base would newly cause intriguing electronic properties including superconductivity. 

The description derived above gives useful insight into the general characteristics of the band structure in solids. In reality, band structure is far more complex than suggested by Fig. 6.16, as a result of the inclusion of three dimensions, and due to the presence of many types of orbitals that form bands. The detailed electronic structure determines the physical and chemical properties of the solids, in particular whether a solid is a conductor, semiconductor, or insulator (Fig. 6.17). 

Accepting that the electronic structure of the metal clusters is in between the discreet electronic levels of the isolated atoms and the band structure of the metals, it is expectable that under a certain size the particle becomes nonmetallic. Indeed, theoretical estimations [102,105] suggest that the gap between the filled and empty electron states becomes comparable with the energy of the thermal excitations in clusters smaller than 50-100 atoms or 1 nm in size, where the particles start to behave as insulators. A... 

As to the number of atoms required to close the gap between insulator and metallic clusters, they vary from as few as 20 to several hundred atoms. Freund3 suggests that the precise numbers will vary from metal to metal, depending on the electronic structure of the metal. 

The competition between these two terms produces a large variety of electronic structures in molecular systems. The condition l U favors itinerant metallic states, whereas the condition t stabilizes localized insulating states. In the latter case, the Hubbard Hamiltonian is reduced to the Heisenberg Hamiltonian... 

The decomposition of nitrous oxide over various metal oxides has been widely investigated by many investigators (1-3). Dell, Stone and Tiley (4) have compared the reactivity of metal oxides and shown that in general p-type oxides were the best catalysts and n-type the worst, with insulators occupying an intermediate position. It has been generally accepted (5) that this correlation indicates that the electronic structure of the catalyst is an important factor in the mechanism of the decomposition of nitrous oxide over metal oxides catalysts. The reaction is usually written (4) as... 

We shall briefly discuss the electrical properties of the metal oxides. Thermal conductivity, electrical conductivity, the Seebeck effect, and the Hall effect are some of the electron transport properties of solids that characterize the nature of the charge carriers. On the basis of electrical properties, the solid materials may be classified into metals, semiconductors, and insulators as shown in Figure 2.1. The range of electronic structures of oxides is very wide and hence they can be classified into two categories, nontransition metal oxides and transition metal oxides. In nontransition metal oxides, the cation valence orbitals are of s or p type, whereas the cation valence orbitals are of d type in transition metal oxides. A useful starting point in describing the structures of the metal oxides is the ionic model.5 Ionic crystals are formed between highly electropositive... 

In this chapter, we have discussed the application of metal oxides as catalysts. Metal oxides display a wide range of properties, from metallic to semiconductor to insulator. Because of the compositional variability and more localized electronic structures than metals, the presence of defects (such as comers, kinks, steps, and coordinatively unsaturated sites) play a very important role in oxide surface chemistry and hence in catalysis. As described, the catalytic reactions also depend on the surface crystallographic structure. The catalytic properties of the oxide surfaces can be explained in terms of Lewis acidity and basicity. The electronegative oxygen atoms accumulate electrons and act as Lewis bases while the metal cations act as Lewis acids. The important applications of metal oxides as catalysts are in processes such as selective oxidation, hydrogenation, oxidative dehydrogenation, and dehydrochlorination and destructive adsorption of chlorocarbons. 

Next, we consider the electronic structure of a metal formed from atoms each contributing two electrons. We have seen that overlap of v orbitals in N atoms produces A/ molecular orbitals and that each orbital can accommodate two electrons. The maximum number of electrons that can be placed in N orbitals is 2N, When each atom contributes two electrons, there are 2A/ electrons to be placed in molecular orbitals. Thus, when each atom contributes two electrons, the band is full and the material is an insulator (Fig. 3,12b). The major success of band theory rests on the explanation of the three types of electrical conductors (Fig. 3.12). 

In previous chapters we focused on physical properties for which electrons are only important in the sense that we must know the ground state of the electrons to understand the material s energy. There is, of course, a long list of physical properties where the details of the electronic structure in the material are of great importance. Two examples of these properties are the classification of a bulk material as a metal, a semiconductor, or an insulator and the existence and characteristics of magnetic properties. In this chapter we examine how information related to these questions can be obtained from DFT calculations. 

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