Electronic structure of crystalline solid

When studying the electronic structure of crystalline solids, physicists tend to think in terms of the mathematical concept of electronic bands, the so-called dispersion... 

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... 

Henderson, Eds., Academic Press, New York, 1975, p. 147. Hartree-Fock Studies of Electronic Structures of Crystalline Solids. 

A general presentation and discussion of the origin of structure of crystalline solids and of the structural stability of compounds and solid solutions was given by Villars (1995) and Pettifor (1995). For an introduction to the electronic structure of extended systems, see Hoffmann (1987, 1988). In this chapter a brief sampling of some useful semi-empirical correlations and, respectively, of methods of classifying (predicting) phase and structure formation will be summarized. 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

The band structure of crystalline solids is usually obtained by solving the Schrodinger equation of an approximate one-electron problem. In the case of non-metallic materials, such as semiconductors and insulators, there are essentially no free electrons. This problem is taken care of by the Bloch theorem. This important theorem assumes a potential energy profile V(r) being periodic with the periodicity of the lattice. In this case the Schrodinger equation is given by... 

Eshelby, J. D. (1954). Distortion of a Crystal by Point Imperfections. Journal of Applied Physics, Vol. 25, No. 2, (February 1954), pp. 255-261. ISSN 0021-8979 Eshelby, J. D. (1956), The continuum theory of lattice defects. Solid State Physics Advances in Research and Applications. Frederick Seitz and David Turnbull, (Ed.), Vol. 3, (1956), pp. 79-144, Elsevier, ISBN 978-0-12-374292-6, Amsterdam, the Netherlands Friedel, J. (1954). Electronic structure of primary solid solutions in metals. Advances in Physics, Vol. 3, No. 12, (October 1954), pp. 446-507, ISSN 0001-8732 Fan, G. J. Choo, H. Liaw, P. K. (2007). A new criterion for the glass-forming ability of liquids. Journal of Non-Crystalline Solids, Vol. 353, No.l, (January 2007), pp. 102-107, ISSN 0022-3093... 

The band structure of crystalline solids is usually obtained by solving the Schrodinger equation of an approximate one-electron problem. In the case of... 

The most extensive calculations of the electronic structure of fullerenes so far have been done for Ceo- Representative results for the energy levels of the free Ceo molecule are shown in Fig. 5(a) [60]. Because of the molecular nature of solid C o, the electronic structure for the solid phase is expected to be closely related to that of the free molecule [61]. An LDA calculation for the crystalline phase is shown in Fig. 5(b) for the energy bands derived from the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for Cgo, and the band gap between the LUMO and HOMO-derived energy bands is shown on the figure. The LDA calculations are one-electron treatments which tend to underestimate the actual bandgap. Nevertheless, such calculations are widely used in the fullerene literature to provide physical insights about many of the physical properties. 

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

To answer this question we need to consider the kind of physical techniques that are used to study the solid state. The main ones are based on diffraction, which may be of electrons, neutrons or X-rays (Moore, 1972 Franks, 1983). In all cases exposure of a crystalline solid to a beam of the particular type gives rise to a well-defined diffraction pattern, which by appropriate mathematical techniques can be interpreted to give information about the structure of the solid. When a liquid such as water is exposed to X-rays, electrons or neutrons, diffraction patterns are produced, though they have much less regularity and detail it is also more difficult to interpret them than for solids. Such results are taken to show that liquids do, in fact, have some kind of long-range order which can justifiably be referred to as a structure . 

Using solid-state physics and physical metallurgy concepts, advanced non-destructive electronic tools can be developed to rapidly characterize material properties. Non-destructive tools operate at the electronic level, therefore assessing the electronic structure of the material and any perturbations in the structure due to crystallinity, defects, microstructural phases and their features, manufacturing and processing, and service-induced strains.1 Electronic, magnetic, and elastic properties have all been correlated to fundamental properties of materials.2 5 An analysis of the relationship of physics to properties can be found in Olson et al.1... 

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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