The Electronic Structure of Crystalline Solids

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... 

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... 

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... 

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... 

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. 

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... 

In the case of metal clusters, for example, valence electrons show the shell structure which is characteristic of the system consisting of a finite number of fermions confined in a spherical potential well [2]. This electronic shell structure, in turn, motivated some theorists to study clusters as atomlike building blocks of materials [3]. The electronic structure of the metallofullerenes La C60 [4] and K C60 [5] was investigated from this viewpoint. This theorists dream of using clusters as atomlike building blocks was first realized by the macroscopic production of C60 and simultaneous discovery of crystalline solid C60, where C60 fullerenes form a close-packed crystalline lattice [6]. 

A unique method for studying the phase composition and the atomic structure of crystalline materials. -> Electrodes and -> solid electrolytes are usually crystalline materials with a regular atomic structure that predicts their electrochemical behavior. For instance, the ionic transport in solid electrolytes or - insertion electrodes is possible only owing to the special atomic arrangement in these materials. The method is based on the X-ray (neutron or electron) reflection from the atomic planes. The reflection angle 9 depends on the X-ray (neutron or electron) wave length A and the distance d between the atomic planes (Braggs Law) ... 

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 electrical conductivity of a material is a macroscopic solid-state property since even in high molecular-weight polymers there is not just one conjugated chain which spans the distance between two electrodes. Then it is not valid to describe the conductivity by the electronic structure of a single chain only, because intra- and interchain charge transport are important. As with crystalline materials, some basic features of the microscopic charge-transport mechanism can be inferred from conductivity measurements [83]. The specific conductivity a can be measured as the resistance R of a piece of material with length d and cross section F within a closed electrical circuit,... 

Crystalline and amorphous silicons, which are currently investigated in the field of solid-state physics, are still considered as unrelated to polysilanes and related macromolecules, which are studied in the field of organosilicon chemistry. A new idea proposed in this chapter is that these materials are related and can be understood in terms of the dimensional hierarchy of silicon-backbone materials. The electronic structures of one-dimensional polymers (polysilanes) are discussed. The effects of side groups and conformations were calculated theoretically and are discussed in the light of such experimental data as UV absorption, photoluminescence, and UV photospectroscopy (UPS) measurements. Finally, future directions in the development of silicon-based polymers are indicated on the basis of some novel efforts to extend silicon-based polymers to high-dimensional polymers, one-dimensional superlattices, and metallic polymers with alternating double bonds. 

Although interesting within the framework of polymer physics and material science this would not be sufficient to attract so many workers from areas outside of conventional polymer research. Additional interest arouse because of the unusual structure of the polymers obtained via solid-state polymerization of diacetylenes and because of the mechanistic features related to its formation. Polydiacetylenes exhibit a fully conjugated and planar backbone in the crystalline state and are thus considered the prototype study object as far as the nature and physical behavior of polyconjugated macromolecules are concerned Theoretical discussions of the electronic structure of these polymers (2) lead to a description in terms of a wide band one-dimensional semiconductor... 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

The chemical elements provide examples of three of the four classes of crystalline solids described in this section. Only ionic solids are excluded, because a single element cannot have the two types of atoms of different electronegativities needed to form an ionic material. We have already discussed some of the structures formed by metallic elements, which are sufficiently electropositive that their atoms readily give up electrons to form the electron sea of metallic bonding. The nonmetallic elements are more complex in their structures, reflecting a competition between intermolecular and intramolecular bonding and producing molecular or covalent solids with varied properties. 

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