Band theory

The band theory is based on the fact that each orbital of the atom represents a single energy level, but when joined, lose their identity and together form electronic bands. Thus, for example, the corresponding energy levels S and p as function of the interatomic distance is almost constant, but as they approach, they lose their identities and form bands, as shown below [7]. 

One observes that until a certain interatomic distance, the energy levels are independent, but from a certain distance, the electrons coexist, forming the band which increases with decreasing interatomic distance (Fig. 5.5a). 

The electronic density N(E) represents the number of layers or levels with electrons with energy between E and E + AE, which are transferred from one level to another until obtaining the maximum energy level, known as Fermi energy p (Fig. 5.5b). 

This density N(E) increases with /E [7]. The electrons always occupy the lower levels. In general there are two bands. First, the conduction band sp. This band extends through the metal and conducts electricity. The localized bands are called d bands, which are closer to the atoms. They are usually represented by the valence band. Secondly, the valence band is the band with the highest electron occupancy level and the conduction band goes to the highest level. The electrons of the d band 

There are two ways to interpret the bands in the structures. First, considering the Fermi level E-p that represents the chemical potential of the electrons in the band. Below the Fermi level, the band is occupied by electrons, and above this level the band is completely empty. Second, through the electron density of the band per unit energy, because the electron distribution is not homogeneous. 

This happens in crystals where many atoms are bonded. There can exist a gap in the band that represents the energy difference between electrons in the valence band with respect to those of the conduction band. The size of the band gap can have important implications for use in optics and electronics. For more about band gaps, see Chapter 18. 

A simple model to describe metallic bonds is the so-called electron sea model. Metals can be considered as metal cations surrounded by valence electrons that swim around in all directions like in a sea. In that way, metals have high electrical and thermal conductivity in all directions since the valence electrons freely can move around. In order to describe this in more details we have to introduce the so-called band theory. In the band theory the molecular orbitals (that we heard about in the section 2.2.2 Molecular orbital theory) are again included. 

The large amount of atomic orbitals constitutes in practice a continuous energy transition between the molecular orbitals. This is the energy band. The bond orbitals in lithium metal will be occupied each with one electron and the anti-bond orbitals are empty. Because the transition from bond orbital to anti-bond orbital is very small in terms of energy, the electrons can easily move from at bond orbital to anti-bond orbital. Thus is easy to get a current of electrons transported through the metallic structure because the electrons can easily move in the empty anti-bond orbitals. They can flow through the metal as an electron sea. This is thus an explanation of the high metallic electrical conductance in all directions from the very 

In a crystal, the wavefunctions and density are formally of infinite extent and so to represent the wavefunctions using the linear combination of atomic orbitals familiar from molecular quantum mechanics would require an infinite basis set. Such an approach is clearly impractical however, the periodicity of the system suggests that only the unit cell of the lathee is really required. This, with some provisos ouhined below, is indeed the case. A very comprehensive discussion of band theory and the implementation of periodic DFT has been published by Payne and coworkers [25], and more recently Hill and coworkers have produced a book covering background theory and applications in materials science [26]. In this sechon we give an overview of the methodology that takes the molecular orbitals of isolated species into the band states of solids. 

Firstly, note that it is the electron density that is observed experimentally in X-ray diffrachon (XRD) experiments and so is used to define the unit cells of crystalline solids. In a periodic system, this means that the electron density has to have the same repeat distance as the lathee in all directions. Bloch s theorem points out that the restriction this imposes on the underlying wavefunchons is actually less rigorous since the relationship between electron density, p(r), at an arbitrary point, r, and the one-electron wavefunctions obtained from calculations (HF or DFT) is  

N is the number of cells in the sample along the a-direchon with N), and the number along b and c respechvely, I, m and n are integers with ranges set by the corresponding N value, -Na/2 I N /2 and so on. In Expression 8.18 reciprocal 

Multipliers for a, h (A) Formula of repeating unit Cell volume relative to Rhomb, cell Binding energy per unit volume (eVA 

The rhombohedral unit cell is the primitive cell all other calculations employ the more usual 

Although metals are generally good conductors of electricity, there is still some resistance to electrical flow, which is known as the resistivity of the metal. At normal temperatures, the resistivity is caused by the flow of electrons being impeded because of the motion of atoms that results from vibration about mean lattice positions. When the temperature is raised, the vibration of atoms about their mean lattice positions increases in amplitude, which further impedes the flow of electrons. Therefore, the resistivity of metals increases as the temperature increases. In a metal, electrons move throughout the structure. There are usually a small number of electrons from each atom that are considered, and because in most structures (fee and hep) each atom has 12 nearest neighbors, there is no possibility for the formation of the usual bonds that require two electrons for each. As a result, individual bonds are usually weaker than those of ionic or covalent character. Because of the overall number of bonds, the cohesion in metals is quite high. 

As metal atoms interact with nearest neighbors at relatively short distance, orbital overlap results in electron density being shared. As mentioned earlier, that electron density is delocalized in orbitals that are essentially molecular orbitals encompassing all of the atoms. The number of atoms that contribute an orbital to the molecular orbital scheme approaches the number of atoms present. As two atoms 

Therefore, there are N energy levels that span an energy band that approaches 4ft in overall width with the energy separating levels k and k + 1 approaching zero. 

Because the highest occupied band is only half filled, electrons can move into the band and move within the band during electrical conduction. Light in the visible region of the spectrum interacts with 

In the free electron model, the electrons are presumed to be loosely bound to the atoms, making them free to move throughout the metal. The development of this model requires the use of quantum statistics that apply to particles (such as electrons) that have half integral spin. These particles, known as fermions, obey the Pauli exclusion principle. In a metal, the electrons are treated as if they were particles in a three-dimensional box represented by the surfaces of the metal. For such a system when considering a cubic box, the energy of a particle is given by 

The important physical properties of simple metals and, in particular, the alkali metals can be understood in terms of a free electron model in which the most weakly bound electrons of the constituent atoms move freely throughout the volume of the metal (231). This is analogous to the free electron model for conjugated systems (365) where the electrons are assumed to be free to move along the bonds throughout the system under a potential field which is, in a first approximation, constant (the particle-in-a-box model). The free electron approach can be improved by replaeing the constant potential with a periodic potential to represent discrete atoms in the chain (365). This corresponds to the nearly free electron model (231) for treating electrons in a metal. 

For simplicity, let us consider a process of an approach of several atoms that eventually form an united atomic system. When N atoms are brought together the individual eigenfunctions of electrons begin to overlap. A given energy level of the system is split into N distinct energy levels. 

In first approximation, it can be said that electron waves are diffracted by a crystal plane that is located at the boundary of the Brillouin zone. However, only electrons of a specified wavelength undergo the diffraction. Imagine an electron wave of length A that propagates in a direction perpendicular to atom planes. The parallel planes are at a distance a = d from each other. According to the Bragg equation (4.15), the reflection of waves in the direction 0 = n/l occurs on condition 

The first Brillouin zone is of a primary importance because aU the solutions can be completely characterized by their behavior in the first Brillouin zone. Usually a band structure is only plotted in the first Brillouin zone. 

So electrons with wave vectors on or near the boundary of Brillouin zone are diffracted all others ( normal ) are not. We expect that normaT electrons will not feel any diffraction. They obey the relation for the total energy E = hVl 2me). 

The condition k — ko=g, see (4.16), is satisfied at the boundary of the Brillouin zone. For electrons with the k z vectors we must expect major modifications. The energy value of an electron with the fegz wave vector turns out to be split into two quantities. 

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The first reliable energy band theories were based on a powerfiil approximation, call the pseudopotential approximation. Within this approximation, the all-electron potential corresponding to interaction of a valence electron with the iimer, core electrons and the nucleus is replaced by a pseudopotential. The pseudopotential reproduces only the properties of the outer electrons. There are rigorous theorems such as the Phillips-Kleinman cancellation theorem that can be used to justify the pseudopotential model [2, 3, 26]. The Phillips-Kleimnan cancellation theorem states that the orthogonality requirement of the valence states to the core states can be described by an effective repulsive... 

Much of the work done on metal clusters has been focused on the transition from cluster properties to bulk properties as the clusters become larger, e.g. the transition from quantum chemistry to band theory [127]. 

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... 

Anisimov V i, Zaanen J and Andersen O K 1991 Band theory and Mott insuiators Hubbard U instead of Stoner / Phys. Rev. B 44 943... 

Andersen O K 1975 Linear methods in band theory Phys. Rev. B 12 3060... 

Band theory Ban-Flame process Banked memory Banocide Bantu siderosis Barb an Barbased Barbital... 

Color from Color Centers. This mechanism is best approached from band theory, although ligand field theory can also be used. Consider a vacancy, for example a missing CF ion in a KCl crystal produced by irradiation, designated an F-center. An electron can become trapped at the vacancy and this forms a trapped energy level system inside the band gap just as in Figure 18. The electron can produce color by being excited into an absorption band such as the E transition, which is 2.2 eV in KCl and leads to a violet color. In the alkaU haUdes E, = 0.257/where E is in and dis the... 

Baker-Venkataraman synthesis, 2, 76 Balsoxin, 6, 232 Bamicetin isolation, 3, 147 Band theory semiconductors, 1, 355 Ban thine... 

The beginnings of the enormous field of solid-state physics were concisely set out in a fascinating series of recollections by some of the pioneers at a Royal Society Symposium (Mott 1980), with the participation of a number of professional historians of science, and in much greater detail in a large, impressive book by a number of historians (Hoddeson et al. 1992), dealing in depth with such histories as the roots of solid-state physics in the years before quantum mechanics, the quantum theory of metals and band theory, point defects and colour centres, magnetism, mechanical behaviour of solids, semiconductor physics and critical statistical theory. 

The other place to read an authoritative histoi7 of the development of the quantum-mechanical theory of metals and the associated evolution of the band theory of solids is in Chapters 2 and 3 of the book. Out of the Crystal Maze, which is a kind of official history of solid-state physics (Hoddeson et al. 1992). 

The recognition of the existence of semiconductors and their interpretation in terms of band theory will be treated in Chapter 7, Section 7.2.1. Pippard, in his chapter, includes an outline account of the early researches on semiconductors. 

Understanding alloys in terms of electron theory. The band theory of solids had no impact on the thinking of metallurgists until the early 1930s, and the link which was eventually made was entirely due to two remarkable men - William Hume-Rothery in Oxford and Harry Jones in Bristol, the first a chemist by education and the second a mathematical physicist. 

A book edited by Levinson (1981) treated grain-boundary phenomena in electroceramics in depth, including the band theory required to explain the effects. It includes a splendid overview of such phenomena in general by W.D. Kingery, whom we have already met in Chapter I, as well as an overview of varistor developments by the originator, Matsuoka. The book marks a major shift in concern by the community of ceramic researchers, away from topics like porcelain (which is discussed in Chapter 9) Kingery played a major role in bringing this about. 

Materials that are comprised of small fragments of (SN) with organic terminal groups, e.g., ArSsNaAr and ArS5N4Ar (Ar = aryl), are of potential interest as molecular wires in the development of nanoscale technology. Consistent with simple band theory, the energy gap... 

Like Rh and Ir, all three members of this triad have the fee structure predicted by band theory calculations for elements with nearly filled d shells. Also in this region of the periodic table, densities and mps are decreasing with increase in Z across the table thus, although by comparison... 

To summarize we have reproduced the intricate structural properties of the Fe-Co, Fe-Ni and the Fe-Cu alloys by means of LMTO-ASA-CPA theory. We conclude that the phase diagram of especially the Fe-Ni alloys is heavily influenced by short range order effects. The general trend of a bcc-fcc phase transition at lower Fe concentrations is in accordance with simple band Ailing effects from canonical band theory. Due to this the structural stability of the Fe-Co alloys may be understood from VGA and canonical band calculations, since the common band model is appropriate below the Fermi energy for this system. However, for the Fe-Ni and the Fe-Cu system this simple picture breaks down. 

H.L. Davis, in Computational Methods in Band Theory , Marcus, Janak and Williams eds., 183 (Plenum... 

We have carried out impurity calculations for a zinc atom embedded in a copper matrix. We first perform self consistent band theory calculations on pure Cu and Zn on fee lattices with the lattice constant of pure Cu, 6.76 Bohr radii. This yields Fermi energies, self consistent potentials, scattering matrices, and wave functions for both metals. The Green s function for a system with a Zn atom embedded in a Cu matrix... 

J.S. Faulkner and T.P. Beaulac, Multiple-scattering approach to band theory. II Fast band theory , Phys. Rev. B26.-1597 (1982) ... 

D.M.Nicholson and J.S.Faulkner, Apphcations of the quadratic Korringa-Kohn-Rostoker band-theory method , Phys.Rev. B39 8187 (1989). 

In Chapter 9, we considered a simple picture of metallic bonding, the electron-sea model The molecular orbital approach leads to a refinement of this model known as band theory. Here, a crystal of a metal is considered to be one huge molecule. Valence electrons of the metal are fed into delocalized molecular orbitals, formed in the usual way from atomic... 

Orbital Region of space in which there is a high probability of finding an electron within an atom, 141 band theory and, 654—655 hybrid, 186,187 relation to ligand, 418... 

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