Semiconductor, parabolic band

In the analyses of conventional ZB semiconductors, we frequently assume a symmetric parabolic band for the conduction band state, and the Luttinger-Kohn Hamiltonian is used to describe the valence band states. In general, the effective Hamiltonian is derived from a k.p perturbation theory or from the theory of invariants developed by Pikus and Bir. In the latter theory, the operator form of the effective Hamiltonian can easily be constructed from symmetry consideration alone. Within this framework, the lowest two conduction bands and the upper six valence bands are described to the second order of k. The invariant forms of the Hamiltonians are written as follows [26,27] ... 

In the analyses of conventional zincblende (ZB) semiconductors, we frequently assume a parabolic band for the conduction bands, and the 6 x 6 Luttinger-Kohn Hamiltonians are used to describe the upper valence bands [1,2], In treating the valence bands together with the conduction bands on an equal footing, as when estimating the momentum matrix elements, we often make use of the 8 x 8 Kane Hamiltonian [3], However, the form of the Hamiltonians reflects the crystal symmetry, and Kane Hamiltonians are constructed under the condition of cubic symmetry. For wurtzite (WZ) materials, therefore, we must consider hexagonal symmetry in the effective Hamiltonian. Let us consider the 8 x 8 k.p Hamiltonian for WZ structure [4,5],... 

Eg between the valence band and the conduction band. The band structure of a direct II-VI intrinsic semiconductor like CdSe can be represented reasonably well by a parabolic band model like that shown schematically in Fig. 2. Here, k = 7r/ris the wave vector and r is the radial distance from an arbitrary origin in the center of the crystal. The kinetic energy of the electron is proportional to E- and the energy minimum of the conduction band and the maxima of the valence bands occur at k = 0 (corresponding to r = co in a bulk sample). 

The analysis of the shape of the absorption edge of the high-pressure phase (Fig. 13) shows the existence of two spectral ranges with different types of energy dependence on the absorption coefficient. At high values of absorption it follows the empirical Tauc relation [57] in the case of parabolic band edges (Fig. 13(b)), while at smaller absorption a so-called Urbach or exponential absorption tail [58, 59] is observed (Fig. 13(c)). The existence of this kind of absorption edge is normally related to amorphous semiconductors. The optical absorption gap determined from our experiment is 0.6-0.7 eV and it decreases with pressure (see below). The slope of the Urbach tail, which can be considered as a measure of a random microfield [59] is found to be T=2.6 eV at 160 GPa. This is very close to what one would expect for an amorphous phase with a coordination of 2.5 [59]. 

Consider now the charge distribution of ionized centers in the depletion region. For comparison, two cases can illustrate the difference between crystalline and a-Si H. The two cases are for n-type semiconductors, one with a single band of donor levels and the other with a uniform density of donor levels throughout the band gap. These two cases are illustrated in Fig. 2. In these cases, Poisson s equation can be explicitly solved. The solutions yield parabolic bands for the case with a single donor band and an exponential behavior for the continuous uniform-state density. The field dependences of the two cases differ also. For the discrete level, a linear dependence results, whereas an exponential behavior is obtained in the uniform-state case. The most striking difference is in the density of ionized states, which is uniform... 

The Hamiltonian of the electron-photon interaction will be used in a very simplified form taking into account only the simplest band structure of a semiconductor with parabolic electron and hole bands without complications related to heavy and light holes, spin-orbit splitted hole band or with the Dirac model of the band structure in the case of small band gap semiconductors. In the case of simple parabolic band after their size quantization in a spherical symmetry quantum dots the electrons and holes are characterized by envelope wave functions with the quantum numbers I, n, m. An essential simplification of the future calculations is the fact that in the selected simple model the band-to-band transitions under the influence of the electron-photon interaction Hamiltonian take place with the creation of an e-h pair with exactly the same quantum numbers for electron and for hole as follows e l,n,m), h l,n,m). ... 

N. G. Nilsson, Empirical approximation for the Fermi energy in a semiconductor with parabolic bands, Appl. Phys. Lett. 33 (1978) 653-654. 

The other approach for reducing (Ay)igggr is by modifying the density of state distribution />( ). This distribution has, according to Eg. [4], a strong effect on the a parameter. The distribution p(u) is a characteristic of the semiconductor and in the parabolic band approximation is given by... 

The Lorentz number of a semiconductor with a parabolic band, due to elastic scattering of electrons can be written as (Oskotski and Smirnov 1972, Smirnov and Tamarchenko 1977)... 

There are various approximations in literature for intrinsic Auger lifetime for the CCCH process for the case of parabolic bands and nondegenerate semiconductor. The most often met is... 

Another important concept in band theory is that of effective mass. In a semiconductor, most of the charges reside at the edge of the conduction or valence band. Band edges can be approximated to parabolic bands, by analogy with the free electron dispersion law, Eq. (3)... 

Ep = 2 edcv /wto is approximately 21 meV for most of the III-V and II-VI semiconductors, no is the linear refractive index, and Kpi, is a material independent constant (1940 cm (eV) for a two-parabolic-band model). It should be mentioned that the above model does not correctly account for the degeneracy of the valence band (heavy hole, light hole, and spin-orbit/crystal-field split-off bands) and assumes single parabolic conduction and valence bands. Using the Kane band structure with three valence bands and including excitonic effects have been shown to produce larger TPA coefficients [227]. 

Here /ie and are effective masses of electron and hole, respectively. Near to bottom of conductivity band and near to top of valent band where dependence E from k is close to parabolic, electron and hole move under action of a field as particles with effective masses fie — h2l(d2Ec(k)ldk1) and jUh = —h2l( E (k)ldk ) [6]. In particular, in above-considered onedimensional polymer semiconductor /ie — /ih — h2AEQj2PiP2d2 [6]. As a first approximation, it is possible to present nanocrystal as a sphere with radius R, which can be considered as a potential well with infinite walls [6], The value of AE in such nanocrystal is determined by the transition energy between quantum levels of electron and hole, with the account Coulomb interaction between these nanoparticles. 

The electron affinity of a semiconductor relates the vacuum level to the conduction band minimum at the surface. An informative way to view the electron affinity is as the conduction band offset between the semiconductor and vacuum. The band structure of the vacuum is simply the parabolic free electron bands, and the minimum energy (or vacuum level) refers to an electron at rest. For most materials, an electron at the bottom of the conduction band is bound to the material by a potential barrier of several volts. This barrier is the electron affinity and is defined as a positive electron affinity. In some instances, the vacuum level can actually align below the conduction band minimum. This means that an electron at the minimum of the conduction band would not see a surface barrier and could be freely emitted into vacuum. This situation is termed a negative electron affinity. 

In the case of solids, the band dispersion describes a complicated dependence of energy on momentum, that usually cannot be described analytically. However, in the case of a semiconductor, the dispersion relations at the top of the valence band (TVB) and at the bottom of the conduction band (BCB), can often be described approximately as parabolic. Therefore, near the band edges, the delocalized electrons or holes, follow a quadratic equation of the form... 

The effective mass of electron or hole is defined as h2 l d2E k) / dk2. Near the bottom of the band where the E k) versus k curve is close to parabolic, the effective mass (m ) is + P2)IPiP2a2- Therefore, the curvature of the band and the effective mass of the electron or hole are determined by p. With large p (in a strongly interacting system such as inorganic semiconductors), the curvature of the band is steep, the electron or hole effective mass is small, and the bandwidth is large a more pronounced quantum size... 

Equation (5) is simple and widely used. It has the basic physics of the quantum size effect. However, it also has two major deficiencies (1) For most of the semiconductor material, the energy band is not parabolic and the effective mass approximation is not valid except near the k = 0 region. This is especially true for small clusters and small band-gap material. (2)... 

The E0 transition in GaAs is the simplest single transition that can be investigated within the III/V materials and a detailed quantitative fit has been attempted to the data of Fig. 29. A careful analysis of these data lead to the fit shown in Fig. 17 and it is clear that all the features of the experimental spectrum can be reproduced with some precision provided that the manufacturer s acceptor density be taken as the basis for the analysis. By comparing the changes in the Franz-Keldysh oscillation near 820 nm with those calculated using the intermediate field model with a presumed parabolic decay of potential inside the semiconductor depletion layer, it is found that some 70 10% of the potential is dropped inside the depletion layer of this n-type material, as can be seen in Fig. 30, and there is no evidence for the phenomenon described in the previous paragraph whereby the band... 

Finally, it may be noted that the basic aim of this paper is not solely to demonstrate the influence of quantum confinement on the photoemission from non-parabolic semiconductors but also to formulate the appropriate electron statistics in the most generalized form. Transport and other phenomena in semiconductors having different band structures and the derivation of the expressions of many important electronic properties are based on the temperature-dependent electron statistics in such materials. 

Figure 2.8 Free charge carriers in a solid have a parabolic dispersion relation ( (k) oc k ). In a semiconductor, the energy bands for free electrons and holes are separated by an energy gap g. In a bulk semiconductor, the states are quasi-continuous, and each point in the energy bands represents an individual state. In a quantum dot (QD), the charges are confined to a small volume. This situation can be described...

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