Typical Band Structures

What follows is not intended to be an encyclopaedia of band structures. It is offered as an introduction to some of the major classes of simple solids whose band structure has been studied, with illustrations taken from the recent literature. For more comprehensive bibliographies the reader is referred to Slater, Dimmock (for recent applications of the APW and related methods), and Cohen and Heine (for recent applications of semiempirical pseudopotential methods). 

Not many compounds or alloys have been included, partly because the possible combinations of elements become too numerous and partly because much remains to be done in that area. The calculated band structures of these systems often depend critically on assumptions concerning charge transfer (see Section 6.4). It remains to be seen whether the very great successes of band theory for the pure 

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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. 

The fuzzy frontier between the molecular and the nanometric level can be elucidated from an electronic point of view. Molecules and small clusters can be described as systems in which the metal atoms form well-defined bonding and antibonding orbitals. Large clusters or small nanoparticles (quantum dots) with dimensions of a few nanometers are intermediate between the size of molecules and bulk material, presenting discrete energy levels with a small band gap owing to quantum-mechanical rules. Finally, larger particles tend to lose this trend and display a typical band structure similar to that of the bulk material. 

Let us examine present ideas on the reasons for the accumulation of iron in the Precambrian, on the sources and forms of transport of the iron and silica, and on the method of deposition and mechanism of formation of the typical banded structure. 

Basically, when analysing the band structures, the equivalent observations apply to typical solid state compounds like thallium halides and lead chalcogenides. In studies on the origin of distortion in a-PbO, it was found that the classical theory of hybridization of the lead 6s and 6p orbitals is incorrect and that the lone pair is the result of the lead-oxygen interaction [44]. It was also noted... 

B3N6] A [BN3] and N (Fig. 8.11). Band-structure calculations performed for La3(B3N5) revealed a band gap in the order of 4 eV. The corresponding nitridoborate oxide La5(BN3)Og [30] is also salt-Hke, owing the typical nitridoborate structure pattern regarding the environment of the [BN3] ion with lanthanum... 

Si(Pc)0] (S04)o.09)n> i-s limited by the oxidative stability of the sulfate anion. Thermoelectric power, optical reflectivity, magnetic susceptibility, and four-probe electrical conductivity measurements evidence behavior typical of an [Si(PcP+)0]n compound where p 0.20. That is, there is no evidence that the more concentrated counterion charge has induced significant localization of the band structure. 

In a supercell geometry, which seems to have become the method of choice these days, the impurity is surrounded by a finite number of semiconductor atoms, and what whole structure is periodically repeated (e.g., Pickett et al., 1979 Van de Walle et al., 1989). This allows the use of various techniques that require translational periodicity of the system. Provided the impurities are sufficiently well separated, properties of a single isolated impurity can be derived. Supercells containing 16 or 32 atoms have typically been found to be sufficient for such purposes (Van de Walle et al., 1989). The band structure of the host crystal is well described. 

The experimental weight loss in the first and second step (4.0 and 4.5% respectively) is in agreement with that corresponding to condensation to pyrophosphate (4.0%) and polyphosphate (4.2X, n 1). Furthermore, the IR spectra of melamine phosphate and of the residues at 300 and 330 C (Figure 14 spectra A, B and C respectively) show that besides the typical bands of phosphate salts (950-1300 cm-1) which are present in the three spectra, a new absorption due to P-O-P bonds (ca. 890 cm-1) appears in the spectra of the residues. The absorptions due to melamine salt structures (e.g. 780-790 and 1450-1750 cm ) are closely similar in the three spectra of Figure 14. Fire retardants based on melamine pyrophosphate and polyphosphate are reported in the literature 51 as well as methods for preparation of these salts (25-... 

Fein, A.P. J. Vac. Sci. Technol. A. in press)(3). Electronic structure measurements of occupied states are typically made with UPS, while unoccupied states are probed by IPS (49). EELS probes both filled and unfilled states simultaneously, and is therefore used in conjunction with either UPS or IPS to complete a band structure determination (44,49). A new electronic spectroscopy technique, Field Emission Scanning Auger Microscopy (50), utilizes STM-like technology to effect highly localized (c.a. 1 /im) Auger electron spectroscopy. The local electronic information afforded by STM is a valuable complement to these other techniques, and STM is the only one of these methods that may be applied to in situ investigations in condensed media. 

Figure 12.17. (a) Diode laser band structure. (1) In thermal equilibrium. (2) Under forward bias and high carrier injection. Ec, v, and f are the conduction band, valence band, and Fermi energies respectively, (b) Fabry-Perot cavity configuration fora GaAs diode laser. Typical cavity length is 300//m and width 10/tm. d is the depletion layer. 

The occurrence of bipolaronic states in the polymer chains promotes optical absorption prior to the n-n gap transitions. In fact, referring to the example (9.30) of the band structure of doped heterocyclic polymers, transitions may occur from the valence band to the bipolaronic levels. These intergap transitions are revealed by changes in the optical absorptions, as shown by Fig. 9.8 which illustrates the typical case of the spectral evolution of polydithienothiophene upon electrochemical doping (Danieli et al., 1985). 

Conjugated conducting polymers consist of a backbone of resonance-stabilized aromatic molecules. Most frequently, the charged and typically planar oxidized form possesses a delocalized -electron band structure and is doped with counteranions (p-doping). The band gap (defined as the onset of the tt-tt transition) between the valence band and the conduction band is considered responsible for the intrinsic optical properties. Investigations of the mechanism have revealed that the charge transport is based on the formation of radical cations delocalized over several monomer units, called polarons [27]. 

The band structure of a semiconductor is often plotted with the total energy of an electron as the ordinate and the distance through the crystal as the abscissa. A typical example is shown in Fig. 1. Energy levels... 

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