Band gap in solids

It seems inevitable that the electronic chemical potential will play an important role in chemistry for some decades, at least The hardness, or rate of change of p with changes in p or N, also seems destined to be important The identification of T with the HOMO-LUMO gap (or the band gap in solids) also suggests many future uses for the concept of hardness. Whether the word hardness will survive to describe these quantities remains to be seen, but it seems to be appropriate. 

The KS orbital energies can also be used for qualitative interpretation of the electronic spectra of atoms and molecules " and band gaps in solids.In the HF or semiempirical ZDO methods, the unoccupied MOs are subject to the self-consistent field of all N electrons, whereas the occupied MOs are subject to the self-consistent field of the N - 1) electrons (an electron in an occupied orbital does not interact with itself). So, for unoccupied orbitals of the N-electron system, the MO energy Sa corresponds to the interactions of an extra N + l)th electron. For the excitation of an electron from the occupied MO to the unoccupied MO an electron in electron affinity, namely with , . As a result, the MO energy differences, Ea — Si, obtained from HF or semiempirical INDO/S calculations are not estimates of transition energies, they have to be combined with appropriate Jand Kintegrals (see Chapter 2.38). 

The most conspicuous feature of the level diagram is probably the contrast in the position of the VBM and CBM of liquid water. The GGA is notorious for underestimating band gaps in solids. The discrepancy becomes worse for larger band gaps. Our results for water are in line with this expectation. The Kohn-Sham band gap... 

The concept of chemical hardness was originally developed as a measure of the stability of molecules. Its relationship to physical hardness and to solids is discussed here. Also, it is pointed out that shear moduli and polarizabilitites, as well as band gaps in covalent crystals, are related to it. 

The energy levels described in the previous section must be viewed in the context of the solid surrounding the defects. The main energy landscape in a solid is the band structure (Supplementary Material S2). In the simplest depictions, the upper energy band (the conduction band) is separated from the lower energy band (the valence band) by a constant band gap. In real structures, the band architecture is more complex. 

The F center absorption maximum for KC1 is at 565 nm and that for KF is 460 nm (Table 9.1). (a) What is the composition of a natural crystal with color centers showing an absorption peak at 500 nm (b) If the absorption peak for KF corresponds to the promotion of an electron from the F center to the conduction band, determine the energy of the color center with respect to the conduction band. (The band gap in KF is 10.7 eV.) If the relative position of the color center energy level remains the same throughout the KF-KC1 solid solution range, estimate (c) the band gap of KC1 and (d) the band gap for the natural crystal. 

Many other applications of multi-photon absorption spectroscopy have meanwhile been reported in photochemistry and also in solid state physics, for instance, a new assignment of the band gap in alkali bromides by Froehlich et al Some further examples will be discussed in Section 111.10). 

Electron correlation plays an important role in determining the electronic structures of many solids. Hubbard (1963) treated the correlation problem in terms of the parameter, U. Figure 6.2 shows how U varies with the band-width W, resulting in the overlap of the upper and lower Hubbard states (or in the disappearance of the band gap). In NiO, there is a splitting between the upper and lower Hubbard bands since IV relative values of U and W determine the electronic structure of transition-metal compounds. Unfortunately, it is difficult to obtain reliable values of U. The Hubbard model takes into account only the d orbitals of the transition metal (single band model). One has to include the mixing of the oxygen p and metal d orbitals in a more realistic treatment. It would also be necessary to take into account the presence of mixed-valence of a metal (e.g. Cu ", Cu ). 

Fig. 7.16 Schematic illustration of the opening-up of the hybridization band gap in the energy bands of tetrahedral sp-valent solids once the strength of the bond integral becomes sufficiently large. The positions of C, Si, Ge, and Sn along the horizontal axis are marked according to the relative values of their experimental band gaps. (After Cox (1991).)...

It is the Peierl s instability that is believed to be responsible for the fact that most CPs in their neutral state are insulators or, at best, weak semiconductors. Hence, there is enough of an energy separation between the conduction and valence bands that thermal energy alone is insufficient to excite electrons across the band gap. To explain the conductive properties of these polymers, several concepts from band theory and solid state physics have been adopted. For electrical conductivity to occur, an electron must have a vacant place (a hole) to move to and occupy. When bands are completely filled or empty, conduction can not occur. Metals are highly conductive because they possess unfilled bands. Semiconductors possess an energy gap small enough that thermal excitation of electrons from the valence to the conduction bands is sufficient for conductivity however, the band gap in insulators is too large for thermal excitation of an electron accross the band gap. 

UV-visible spectral measurements show that substituted indolo[ 3,2-ib] carbazoles (5) absorb significantly below 450 nm (Fig. 4.12). Their band gaps in the solid state, estimated from the on-set UV-visible absorption, were >2.55 eV, which are significantly larger than those of most p-channel organic semiconductors for OTFTs (Table 4.1). In addition, whereas chlorine substitution at the 3,9 positions of 5,ll-dialkylindolo[3,2-jb]carbazole (i.e. 5d) did not cause noticeable changes in the spectral properties, substitution at the 2,8 positions (i.e. 5c) results in red-shifts in both solution and thin film absorption - a phenomenon that clearly indicates the pronounced electronic effects of substituents at positions para to the radical cation sites (5-N and 11-N positions). 

The electronic properties of solid materials are normally described in terms of the band gap. In this description, the valence band is the ground state and the... 

The identification of the band gap in ionic crystals with pscudopotentials suggests one other property that may be attributable to ionic crystals. The general insensitivity of to material or structure gives a rationalization to the observation that band gaps in ionic solids and even inert-gas solids vary as cl. However, the point cannot be made as strongly for ionic solids as it can for covalent solids. 

An aluminum antimonide solid-state laser emits light with a wavelength of 730. nm. Calculate the band gap in joules. 

The electrical property of a material is determined by its electronic structure, and the relevant theory that explains the electronic structure in the solid state is band theory. This theory, however, does not fully explain conductivity in polymers. It is noteworthy that the energy spacing between the highest occupied and lowest unoccupied bands is called the bandgap the highest occupied band is called the valence band and the lowest unoccupied band is called the conduction band. The bandgaps of insulators and semiconductors are wide and narrow, respectively there are no band-gaps in metals. 

One factor affecting the dielectric strength is the electronic structure of the polymer, and in particular its band gap. In quantum mechanics [29], each electron in a molecule can only occupy one of a discrete set of allowed energy levels. In solids, the overlaps between different repeating units of the material (for example, the repeat units in quasi-one-dimensional systems such as polymer chains [29-31]) cause these discrete energy levels to broaden into bands. The band gap is the energy difference between the top of the valence band and the bottom of the conduction band. (In terms which are equivalent but more familiar to chemists, the band gap is... 

The photoelectrical behaviour of an illuminated solid depends not only on its optical properties but also on the nature of the contacts made to it. If a resistive material is provided with ohmic contacts, there is no barrier to the transfer of electrons to and from the solid and the current that flows when a potential difference is applied between the contacts depends only on the density and mobility of charge carriers. The current will increase if illumination raises the free carrier density considerably and the material is then termed a photoconductor. In practice, illumination may serve either to promote electrons from the valence band to the conduction band or to release carriers trapped at impurity states in the band gap. In both cases, the light gives rise to a volume photoeffect [6]. 

The next three subsections address the not-so-transparent concept of how and why bands form in solids. Three approaches are discussed. The first is a simple qualitative model. The second is slightly more quantitative and sheds some light on the relationship between the properties of the atoms making up a solid and its band gap. The last model is included because it is physically the most tangible and because it relates the formation of bands to the total internal reflection of electrons by the periodically arranged atoms. 

The sp hybridization results in the formation of four energetically degenerate bonds arranged tetrahedrally with each containing one electron. This allows an atom to bond to four other atoms simultaneously. The interactions and overlap of the wave functions of many atoms or ions in a solid give rise to energy bands. If the outermost bands are not filled, the electrons are said to be delocalized and the solid is considered to be a metal. If the bands are separated from each other by a band gap, the solid is considered a semiconductor or insulator depending on the size of that gap. 

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