Band structure in solids

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). 

One area that takes advantage of many of the above formalisms is the application of HF theory to periodic solids. Periodic HF theory has been most extensively developed within the context of the crystal code (Dovesi et al. 2000) where it is available in RHF, UHF, and ROHF forms. Such calculations can be particularly useful for elucidating band structure in solids, assessing defect effects, etc. 

The introduction continues with a brief account of the approximations usually made to arrive at a solvable electronic-structure problem, and we shall discuss several of the methods applied to calculate band structures in solids. Then there is a brief summary of the history and development of the linear methods of band-structure calculations, followed by an outline of the remaining chapters. 

Figure 28.3 Band structure in solids, (a) Sodium, (b) Calcium, (c) Diamond.

Figure 18.4 The various possible electron band structures in solids at 0 K. (a) The electron band structure found in metals such as copper, in which there are available electron states above and adjacent to filled states, in the same band, b) The electron band structure of metals such as magnesium, in which there is an overlap of filled and empty outer bands, (c) The electron band structure characteristic of insulators the filled valence band is separated from the empty conduction band by a relatively large band gap (>2 eV). d) The electron band structure found in the semiconductors, which is the same as for insulators except that the band gap is relatively narrow (<2 eV).

Because of the inverse relationship between interatomic distances and the directions in which constructive interference between the scattered electrons occurs, the separation between LEED spots is large when interatomic distances are small and vice versa the LEED pattern has the same form as the so-called reciprocal lattice. This concept plays an important role in the interpretation of diffraction experiments as well as in understanding the electronic or vibrational band structure of solids. In two dimensions the construction of the reciprocal lattice is simple. If a surface lattice is characterized by two base vectors a and a2, the reciprocal lattice follows from the definition of the reciprocal lattice vectors a and a2 ... 

The simple energy-gap scheme of Figure 4.6 seems to indicate that transitions in solids should be broader than in atoms, but still centered on defined energies. However, interband transitions usually display a complicated spectral shape. This is due to the typical band structure of solids, because of the dependence of the band energy E on the wave vector k ( k =2nl a, a being an interatomic distance) of electrons in the crystal. 

Figure 6.1 Schematic band structures of solids (a) insulator (kT ,) (b) intrinsic semiconductor (kT ,) (c) and (d) extrinsic semiconductors donor and acceptor levels in n-type and p-type semiconductors respectively are shown, (e) compensated semiconductor (f) metal (g) semimetal top of the valence band lies above the bottom of the conduction band.

Interfacial electron transfer across a solid-liquid junction can be driven by photoexcitation of doped semiconductors as single crystals, as polycrystalline masses, as powders, or as colloids. The band structure in semiconductors (281) makes them useful in photoelectrochemical cells. The principles involved in rendering such materials effective redox catalysts have been discussed extensively (282), and will be treated here only briefly. 

Figure 1. Band Structure in a n-type Semiconductor A. Solid State. B. In contact with a liquid phase redox couple (0/R). IL=energy of the conduction band. Vertical line indicates solid-liquid interface. CB= conduction band VB = valence band.

Hagstrom, S. B. M. Photoelectron spectroscopy to study the electronic band structure of solids. In Conference abstracts X-ray photoelectron spectroscopy . Zurich, October 1971. 

We collect here some simple treatments of arrays of atoms or particles in one dimension these will be quite useful in later analogies with the band structure of solids. In particular, the notions of Brillouin30 zone, of band edges, of... 

Work out the band structure for the px, py and pz functions of a two-dimensional square array of B atoms in the xy plane separated by a distance d. This might be a model for a material made up of weakly interacting sheets or of a very thin film or surface layer of a solid. Hint carry out the operations used to generate the band structure in Exercise 6.9 with p functions rather than s functions. 

The NIR spectra of solid compounds 6-10 were measured [101]. The positions of absorption bands of solid [Np02(Pic)(H20)2] and complex particle [Np02(Pic)] in solution were found similar. It may be concluded that the coordination spheres of Np atom are close in both cases, that is, in solution the picolinate ion coordinates to Np with the formation of chelate ring. Alternatively, significant shifts of absorption band to lower energies were observed in spectra of neptunium nicotinate and isonicotinate as the result of the formation of cation-cation structures in solid 9 and 10. 

The band structure of solids has been studied theoretically by various research groups. In most cases it is rather complex as shown for Si and GaAs in Fig. 1.5. The band structure, E(kf is a function of the three-dimensional wave vector within the Brillouin zone. The latter depends on the crystal structure and corresponds to the unit cell of the reciprocal lattice. One example is the Brillouin zone of a diamond type of crystal structure (C, Si, Ge), as shown in Fig. 1.6. The diamond lattice can also be considered as two penetrating face-centered cubic (f.c.c.) lattices. In the case of silicon, all cell atoms are Si. The main crystal directions, F —> L ([111]), F X ([100]) and F K ([110]), where Tis the center, are indicated in the Brillouin zone by the dashed lines in Fig. 1.6. Crystals of zincblende structure, such as GaAs, can be described in the same way. Here one sublattice consists of Ga atoms and the other of As atoms. The band structure, E(k), is usually plotted along particular directions within the Brillouin zone, for instance from the center Falong the [Hl] and the [HX)] directions as given in Fig. 1.5. 

In the previous section we have seen how metal bond can be described according to the band theory. The valence electrons can freely move through the metal lattice in empty anti-bond orbitals. But how are the single atoms arranged relative to each other We are going to look at the answer to this question in this section. Generally two t q)es of structures in solid compounds can be distinguished ... 

Electronic structures of crystalline solids are mostly calculated on the basis of DFT. In this approach an open-shell system is described by spin polarized electronic band structures, in which the up-spin and down-spin bands are allowed to have different orbital... 

B. Segall, F.S. Ham "The Green s Function Method of Korringa, Kohn and Rostoker for the Calculation of the Electronic Band Structure of Solids", in Methods of Computational Physics, Vol.8, ed. by B. Adler,... 

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