Band theory insulators

Terakura K, Qguchi T, Williams A R and Kubler J 1984 Band theory of insulating transition-metal monoxides Band-structure calculations Phys. Rev. B 30 4734... 

O. K. (1991) Band theory and Mott insulators Hubbard Uinstead of Stoner I. Physical Review B - Condensed Matter, 44, 943-954. 

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). 

Further along the series, we saw that stoichiometric MnO, FeO, CoO, and NiO are insulators. This situation is not easily described by band theory because the d orbitals are now too contracted to overlap much, typical band widths are 1 eV, and the overlap is not sufficient to overcome the localizing influence of interelectronic repulsions. (It is this localization of the d electrons on the atoms that gives rise to the magnetic properties of these compounds that are discussed in Chapter 9.)... 

The above simple picture of solids is not universally true because we have a class of crystalline solids, known as Mott insulators, whose electronic properties radically contradict the elementary band theory. Typical examples of Mott insulators are MnO, CoO and NiO, possessing the rocksalt structure. Here the only states in the vicinity of the Fermi level would be the 3d states. The cation d orbitals in the rocksalt structure would be split into t g and eg sets by the octahedral crystal field of the anions. In the transition-metal monoxides, TiO-NiO (3d -3d% the d levels would be partly filled and hence the simple band theory predicts them to be metallic. The prediction is true in TiO... 

In materials in which a metal-insulator transition takes place the antiferromagnetic insulating state is not the only non-metallic one possible. Thus in V02 and its alloys, which in the metallic state have the rutile structure, at low temperatures the vanadium atoms form pairs along the c-axis and the moments disappear. This gives the possibility of describing the low-temperature phase by normal band theory, but this would certainly be a bad approximation the Hubbard U is still the major term in determining the band gap. One ought to describe each pair by a London-Heitler type of wave function... 

Slater s band-theory treatment of Mott-Hubbard insulators... 

The energy states of gaseous atoms split because of the overlap between electron clouds. Obviously, therefore, atoms must come much closer before the clouds of the core electrons begin to overlap compared with the distance at which the clouds of outer (or valence) electrons overlap (Fig. 6.119). Hence, at the equilibrium interatomic distances, the energy levels of the core electrons (in contrast to the valence electrons) do not show any band structure and therefore will be neglected in the following discussion. This simplified picture of the band theory of solids will now be used to explain the differences in conductivity of metals, semiconductors, and insulators. 

Insulators are not amenable to treatment of catalytic activity in terms of electronic band theory. 

The electrical properties of any material are a result of the material s electronic structure. The presumption that CPs form bands through extensive molecular obital overlap leads to the assumption that their electronic properties can be explained by band theory. With such an approach, the bands and their electronic population are the chief determinants of whether or not a material is conductive. Here, materials are classified as one of three types shown in Scheme 2, being metals, semiconductors, or insulators. Metals are materials that possess partially-filled bands, and this characteristic is the key factor leading to the conductive nature of this class of materials. Semiconductors, on the other hand, have filled (valence bands) and unfilled (conduction bands) bands that are separated by a range of forbidden energies (known as the band gap ). The conduction band can be populated, at the expense of the valence band, by exciting electrons (thermally and/or photochemically) across this band gap. Insulators possess a band structure similar to semiconductors except here the band gap is much larger and inaccessible under the environmental conditions employed. 

It is the Peierl s instability that is believed to be responsible for the fact that most CPs in their neutral state are insulators or, at best, weak semiconductors. Hence, there is enough of an energy separation between the conduction and valence bands that thermal energy alone is insufficient to excite electrons across the band gap. To explain the conductive properties of these polymers, several concepts from band theory and solid state physics have been adopted. For electrical conductivity to occur, an electron must have a vacant place (a hole) to move to and occupy. When bands are completely filled or empty, conduction can not occur. Metals are highly conductive because they possess unfilled bands. Semiconductors possess an energy gap small enough that thermal excitation of electrons from the valence to the conduction bands is sufficient for conductivity however, the band gap in insulators is too large for thermal excitation of an electron accross the band gap. 

Most minerals fall into the class of insulator phosphors. The characteristics of the luminescence are usually defined by the electronic structure of an activator ion as modified by the crystal field of the host crystal structure. Although some energy transfer takes place between nearby ions, appearing as the phenomena of co-activation, luminescence poisons, and activator pair interactions, the overall luminescence process is localized in a "luminescent center" which is typically 2 to 3 nm in radius. From a perspective of band theory, luminescent centers behave as localized states within the forbidden energy gap. 

FIG. 11.9 Schematic representation of the band theory of electronic conduction (A) insulators (B) semi-conductors and (C) conducting materials. The hatched bands are allowed bands, the cross hatched bands are occupied. 

The band theory of solids provides a clear set of criteria for distinguishing between conductors (metals), insulators and semiconductors. As we have seen, a conductor must posses an upper range of allowed levels that are only partially filled with valence electrons. These levels can be within a single band, or they can be the combination of two overlapping bands. A band structure of this type is known as a conduction band. 

CO. For that matter, in regards to predicting the type of electrical behavior, one has to be careful not to place excessive credence on actual electronic structure calculations that invoke the independent electron approximation. One-electron band theory predicts metallic behavior in all of the transition metal monoxides, although it is only observed in the case of TiO The other oxides, NiO, CoO, MnO, FeO, and VO, are aU insulating, despite the fact that the Fermi level falls in a partially hUed band. In the insulating phases, the Coulomb interaction energy is over 4 eV whereas the bandwidths have been found to be approximately 3 eV, that is, U > W. 

About one decade after the development of band theory, two Dutch industrial scientists at the NV Philips Corporation, Jan Hendrik de Boer (1899-1971) (de Boer was later associated with the Technological University, Delft) and Evert Johaimes Willem Verwey (1905-1981), reported that many transition metal oxides, with partially filled bands that band theory predicted to be metallic, were poor conductors and some were even insulating (de Boer and Verwey, 1937). Rudolph Peierls (1907-1995) first pointed out the possible importance of electron correlation in controlling the electrical behavior of these oxides (Peierls, 1937). Electron correlation is the term applied to the interaction between electrons via Coulombs law. 

Mott originally considered an array of monovalent metal ions on a lattice, in which the interatomic distance, d, may be varied. Very small interatomic separations correspond to the condensed crystalline phase. Because the free-electron-Uke bands are half-filled in the case of ions with a single valence electron, one-electron band theory predicts metallic behavior. However, it predicts that the array will be metallic, regardless of the interatomic separation. Clearly, this can t be true given that, in the opposite extreme, isolated atoms are electrically insulating. 

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