Band theory of metals and insulators

A band is a group of MOs, the energy differences between which are so small that the system behaves as if a continuous, non-quantized variation of energy within the band is possible. 

A band gap occurs when there is a significant energy difference between two bands. 

The valence orbital of an Li atom is the 2s atomic orbital, and Fig. 6.12 shows schematic MO diagrams for the formation of species incorporating different numbers of Li atoms (see Section 2.3). If two Li atoms combine, the overlap of the two 2 atomic orbitals leads to the formation of two MOs. If three Li atoms combine, three MOs are 

A band gap occurs when there is a significant energy difference between two bands. The magnitude of a band gap is typically given in electron volts (eV) 1 eV=96.485 kJmoL.  

The fundamental concept of band theory is to consider the energies of the molecular orbitals in an assembly of metal atoms. An MO diagram describing the bonding in a metallic solid is characterized by having groups of MOs (i.e. bands) 

An electrical conductor offers a low resistance (measured in ohms, fi) to the flow of an electrical current (measured in amperes. A). 

The electrical resistivity of a substance measures its resistance to an electrical current (equation 6.3). For a wire of 

All Box 6.2 Stainless steel corrosion resistance by adding chromium 

Stainless steels are examples of alloy steels, i.e. ones that contain a si-block metal in addition to carbon. Stainless steels have a significant content of the alloy metal and are of high commercial value because of their high resistance to corrosion. All contain a minimum of 10.5% (by mass) of chromium and the resistance to corrosion arises from the formation of a thin layer of Cr203 ( 13 000 pm thick) over the surface of the steel. The oxide layer passivates (see Section 10.4) the steel and is self-repairing, i.e. if some of the oxide coating is scratched off, further oxidation of the chromium in the steel necessarily repairs the wound . A further property that makes stainless steels commercially important is that they can be polished to satin or mirror finishes and this is easily appreciated in the ranges of stainless steel cutlery available to the consumer. 

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Before the discovery of antiferromagnetism, it was pointed out that, according to the Wilson theory of metals and insulators (Chapter 1), nickel oxide should be a metal—whereas it is a transparent insulator. The nickel ions should have eight d-electrons, and the only splitting of the d-band to be expected is into the eg and t2g bands, with four and six electrons. Peierls explained in 1938 how this... 

Using the band theory of metals, differentiate between a metal, an insulator and a semiconductor. 

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... 

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. 

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. 

Terakura, K., T. Oguchi, A. R. Williams, and J. Ktlbler (1984b). Band theory of insulating transition-metal monoxides band-structure calculations. Phys. Rev. B30, 4734-47. 

As mentioned, the band theory of solids leads to a clear distinction between metals and insulators metals are associated with systems with partially filled bands, whereas insulators are associated with systems with completely full and empty bands. However, for a system that, according to a delocalized description of electrons, would have partially filled bands, it may be advantageous to keep these electrons locaHzed because in that way electrostatic repulsion between them could be minimized. These kind of localized states are usually known as Mott-Huhhard... 

Next, we smdy the properties of solids and see how the band theory explains the difference between conductors (metals) and insulators. We learn the special properties of semiconductors. (20.3)... 

In Section 6.3, we saw that the ability of metals to conduct heat and electricity can be explained by considering the metal to be an array of positive ions immersed in a sea of highly mobile delocalized valence electrons. In this section, we will use our knowledge of quantum mechanics and molecular orbital theory to develop a more detailed model for the conductivity of metals. The model we will use to study metallic bonding is the band theory of conductivity, so called because it states that delocalized electrons move freely through "bands formed by overlapping molecular orbitals. We will also apply band theory to understand the properties of semiconductors and insulators. 

In contrast, Kulikov (1982) calculates the band structures of LaHj and LaHj, using self-consistent local-density functional theory. He stresses the importance and sensitivity of the choice for the crystal potential, finding incipient overlap between the conduction and valence bands for LaHj. The concept leads to an excitonic insulator phase at low temperatures. The low-temperature phase is semiconducting, and the higher-temperature phase is metallic i.e., the interpretation is opposite from that of Fujimori and Tsuda (1981). 

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). 

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... 

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