Energy-band structure

GaAs is an example of a direct bandgap system. Recall that the momentum of a photon is E/c while the momentum of a phonon isE/vg. Since the velocity of sound Vq is very much less than the velocity of light, photons have energy but virtually no momentum. So, direct transitions from a valence band to a conduction band must be represented by a vertical line in the band diagram. Since the minimum in the conduction band is directly above the peak 

Band structure of Si. Note that the minimum in the conduction band does not coincide with maximum in the valence band. (From CheUkowski, J.R. and Cohen, M.L., Phys. Rev., B 14/2,559,1976. With permission.) 

Photons may induce direct or vertical transitions from any occupied band to a higher energy band that is not completely full. Such transitions are called interband transitions and contribute to the absorption spectra. So even in an indirect bandgap material, valence electrons can be promoted vertically to the conduction band by adsorbing photons of sufficient energy. Once in the conduction band, the hot electrons become thermalized through collisions and will eventually move to the lowest point in the band. 

Electrons may also undergo intraband transitions in which they are promoted to a higher energy within their own band by absorbing a photon. But since such transitions must be assisted by a phonon, they, like other indirect transitions, are not as likely to occur as interband transitions. 

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Other methods for detennining the energy band structure include cellular methods. Green fiinction approaches and augmented plane waves [2, 3]. The choice of which method to use is often dictated by die particular system of interest. Details in applying these methods to condensed matter phases can be found elsewhere (see section B3.2). 

Simple metals like alkalis, or ones with only s and p valence electrons, can often be described by a free electron gas model, whereas transition metals and rare earth metals which have d and f valence electrons camiot. Transition metal and rare earth metals do not have energy band structures which resemble free electron models. The fonned bonds from d and f states often have some strong covalent character. This character strongly modulates the free-electron-like bands. 

Is 2s 2p 3s 3p 3d 4s. If the 3d states were truly core states, then one might expect copper to resemble potassium as its atomic configuration is ls 2s 2p 3s 3p 4s The strong differences between copper and potassium in temis of their chemical properties suggest that the 3d states interact strongly with the valence electrons. This is reflected in the energy band structure of copper (figure Al.3.27). 

The existence of carbon nanotubes with diameters small compared to the de Broglie wavelength has been described by Iijima[l,2,3] and others[4,5]. The energy band structures for carbon nanotubes have been calculated by a number of authors and the results are summarized in this issue by M.S. Dresselhaus, G. Dres-selhaus, and R. Saito. In short, the tubules can be either metallic or semiconducting, depending on the tubule diameter and chirality[6,7,8]. The calculated density of states[8] shows singularities... 

The optical properties of solid Sg have been studied by ab initio MO calculations of the energy band structure [70] but no experimental data for solid Sg are known. 

Otto, P., and A. Sutjianto. 1991. Electron Correlation Effects on the Energy Band Structure of Polyglycine, J. Mol. Struct. (Theochem) 231, 277-282. 

Energy Band Structures of the Cation Radical Salts [(ppy)Au(S-S)]2[anion] (S-S = CgH4Sg or CgH4S602, anion =PFfi or BF4 )... 

Fig. 9 Energy band structures of (a) [(ppy)Au(C8H4S8)]2[PF6] (b) [(ppy)Au(C8H4S806)]2[BF4]. (Reprinted with permission from [35]. Copyright 2008 American Chemical Society)...

V. L. Bonch-Bruevich, Effect of Heavy Doping on the Semiconductor Band Structure Donald Long, Energy Band Structures of Mixed Crystals of III-V Compounds Laura M. Roth and Petros N. Argyres, Magnetic Quantum Effects... 

FIGURE 1.8 Energy band structure of a PLED in the configuration of ITO/MEH-PPV/Ca. 

Fig. 5.3. Energy band-structure diagram (in eV) of Ni/ZnO support and pre-(post-)chemisorbed hydrogen adatom level at e0(e ). VB (shaded) and CB of ZnO are of width 6. Fermi level (e/), which coincides with lower edge of CB, is taken as zero of energy. 6-layer Ni film has 6 localized levels lying between band edges (dashed lines), which just overlap ZnO energy gap. Reprinted from Davison et al (1988) with permission from Elsevier.

Our model of positive atomic cores arranged in a periodic array with valence electrons is shown schematically in Fig. 14.1. The objective is to solve the Schrodinger equation to obtain the electronic wave function ( ) and the electronic energy band structure En( k ) where n labels the energy band and k the crystal wave vector which labels the electronic state. To explore the bonding properties discussed above, a calculation of the electronic charge density... 

The empirical approach [7] was by far the most fruitful first attempt. The idea was to fit a few Fourier coefficients or form factors of the potential. This approach assumed that the pseudopotential could be represented accurately with around three Fourier form factors for each element and that the potential contained both the electron-core and electron-electron interactions. The form factors were generally fit to optical properties. This approach, called the Empirical Pseudopotential Method (EPM), gave [7] extremely accurate energy band structures and wave functions, and applications were made to a large number of solids, especially semiconductors. [8] In fact, it is probably fair to say that the electronic band structure problem and optical properties in the visible and UV for the standard semiconductors was solved in the 1960s and 1970s by the EPM. Before the EPM, even the electronic structure of Si, which was and is the prototype semiconductor, was only partially known. 

At this time, the fastest growing area in the field of nanophysics is in the studies of buckyballs and nanotubes. After the discovery [33] of the Qo molecule, many properties of the molecule and solids formed from the molecule were explored. The doped C6o crystals showed interesting behavior, including superconductivity. [34] The standard model, including the GW quasiparticle theory, was used [35] successfully to explore the energy band structure, and the superconducting properties appear to be consistent with the BCS theory. [36]... 

More recently, the many theoretical models proposed for an understanding of GMR effects may be classified into two types of approaches, one based on RKKY (Rudeman-Kittel-Kasuya-Yoshida)-like schemes and the other on energy-band structure calculations. 

On perfect crystalline surfaces, the unperturbed electronic structure is determined by the energy band structure of the surface Bloch waves. This is a consequence of the two-dimensional translational symmetry of the surface. The presence of the tip breaks the translational symmetry of the surface, and the surface electronic structure of the sample is perturbed. 

Stimulated by a variety of commercial applications in fields such as xerography, solar energy conversion, thin-film active devices, and so forth, international interest in this subject area has increased dramatically since these early reports. The absence of long-range order invalidates the use of simplifying concepts such as the Bloch theorem, the counterpart of which has proved elusive for disordered systems. After more than a decade of concentrated research, there remains no example of an amorphous solid for the energy band structure, and the mode of electronic transport is still a subject for continued controversy. 

There are several papers on the energy-band structure of SrTi03, e.g. Soules et al, (1972). Wolfram (1972) emphasized the two-dimensional character of the Ti d-band (the conduction band), giving a rapid rise of N( ) with . This will facilitate polaron formation. 

Figure 19. Energy band structure for palladium hydride...

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