Common semiconductor energy bands

Having armed ourselves with a more detailed idea of how energy bands develop in semiconductors, we now consider some specific examples. It is helpful to begin with an examination of some experimental data for some representative semiconductors. Table 5.3 lists lattice parameters and energy gaps for selected common semiconductors. Several of the trends listed above are illustrated by these results. 

Details of the calculation and discussion of the band structures may be found in the original paper in which these results were presented by Chelikowski and Cohen [5]. 

The decreasing variation in energy with momentum across the diagram is not, perhaps, very surprising given the discussion in Section 5.1. As the chemical splitting increases, the bonds become increasingly ionic. Ionic bonds are relatively 

The detailed quantum mechanical basis for all of these changes is discussed in great detail in Harrison and many other sources but the reader is cautioned that a solid understanding of the notations of quantum mechanics is required to understand the discussion. 

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Semiconductors are materials that contain a relatively small number of current carriers compared to conductors such as metals. Intrinsic semiconductors are materials in which electrons can be excited across a forbidden zone (bandgap) so that there are carriers in both the valence (holes, p-type) and conduction (electrons, ra-type) bands. The crucial difference between a semiconductor and an insulator is the magnitude of the energy separation between the bands, called the bandgap (Eg). In the majority of useful semiconducting materials this is of the order of 1 eV some common semiconductors are listed in Table 1. 

Once electronic equilibrium is established, the surface and the volume of the semiconductor have a common Fermi level, i.e., the same electrochemical potential (depicted by the horizontal line FF in Fig. 22). However, owing to the bending of the bands the position of the Fermi level in the energy spectrum of the crystal (its position relative to the energy bands) will, generally speaking, depend on the distance from the surface. We shall characterize the position of the Fermi level by its distance from the top of the valence band, denoted by e+. Evidently, + = We intro-... 

Semiconductor electrodes, which have much lower charge carrier densities (1013—1019 carriers/cm3), typically absorb in the infrared but exhibit much lower absorption by charge carriers than metals of comparable film thickness, and frequently show a transparency window in much of the visible spectrum due to a substantial band-gap energy, before absorbing again in the ultraviolet. For example, Sn02 and ZnO, like many common semiconductor electrode materi-... 

On the other hand, many organic compounds have a redox potential at a higher energy than the valence band edge of common semiconductor oxides and, therefore, they can act as electron donors and thus yield a radical cation (Fig. 1), which can further react, for example, with H20, 02 , or 02. 

The driving force of the electron transfer process in the interface is the difference of energy between the levels of the semiconductor and the redox potential of the species close to the particle surface. The thermodynamically possible processes occurring in the interface are represented in Fig. 9 the photogenerated holes give rise to the D -> D + oxidative reaction while the electrons of the conduction band lead to the A -> A reductive process. The most common semiconductors present oxidative valence bands (redox potentials from +1 to + 3.5 V) and moderately reductive conduction bands (+ 0.5 to - 1.5 V) [115]. Thus, in the presence of redox species close or adsorbed to the semiconductor particle and under illumination, simultaneous oxidation and reduction reactions can take place in the semiconductor-solution interface. 

The values of Vm and are key experimental quantities that are used to characterize the physical properties of semiconductor/metal interfaces. If Vbi or b can be determined, then W, Q, E(x), and most of the other important thermodynamic quantities that are relevant to the electrical properties of the semiconductor contact can be readily calculated using the simple equations that have been presented above. Methods to determine these important parameters can be found in the literature. However, it would be useful at this point in the discussion to consider what values of and Vbi are expected theoretically for a given semiconductor/metal interface. By definition, = (/ip.m - at the electrode surface (Figure 4b). Thus, in principle, the barrier height can be predicted if the energies of the semiconductor band edges and the electrochemical potential of the metal can be determined with respect to a common reference energy. 

For instance, the most common orders of magnitude for energy-band gaps are 5.3 eV (511 kj/mol) for diamond-type la, showing its excellent electric insulating properties, and 1.04 eV (i.e., 100 kJ/mol) for pure silicon (Si) monocrystal used as semiconductors and 0.69 eV (i.e., 67 kJ/mol) for pure germanium (Ge) crystals. The first two examples are both intrinsic semiconductors. Moreover, electrical conductivity of materials is strongly temperature dependent. In fact, as the temperature increases, the conductivity of metals decreases, while the electrical conductivity of pure semiconductors and insulators increases. 

It is traditional for quantmn theory of molecular systems (molecular quantum chemistry) to describe the properties of a many-atom system on the grounds of interatomic interactions applying the hnear combination of atomic orbitals (LCAO) approximation in the electronic-structure calculations. The basis of the theory of the electronic structure of solids is the periodicity of the crystalline potential and Bloch-type one-electron states, in the majority of cases approximated by a linear combination of plane waves (LCPW). In a quantmn chemistry of solids the LCAO approach is extended to periodic systems and modified in such a way that the periodicity of the potential is correctly taken into account, but the language traditional for chemistry is used when the interatomic interaction is analyzed to explain the properties of the crystalhne sohds. At first, the quantum chemistry of solids was considered simply as the energy-band theory [2] or the theory of the chemical bond in tetrahedral semiconductors [3]. From the beginning of the 1970s the use of powerful computer codes has become a common practice in molecular quantum chemistry to predict many properties of molecules in the first-principles LCAO calculations. In the condensed-matter studies the accurate description of the system at an atomic scale was much less advanced [4]. 

In heterostructures both the CB electrons and the VB holes will experience extra potential energies such as band offsets. For low-energy Frenkel excitons in a QD, the electrons and holes are confined within the same QD volume (type-I exciton) so that we will have a common potential energy V( R - a ) that confines the exciton, where R is the center of mass of the exciton (see O Eq. 23.19), a denotes the center of the QD. For conunon semiconductor QDs, the radius of the QD, Rqo, is in the order of the exciton Bohr radius Br so that one may neglect the free motion of the center of mass of the electron-hole pair in the QD, that is, K = 0 in O Eq. 23.20. For the ground state of the exciton we can approximate V( B-a ) = 0 when li-a < Rqd and y( U-a ) = oo elsewhere so that the wave function corresponding to the motion of the center of mass becomes (Gasiorowicz 1996)... 

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