Semiconductor, parabolic band model

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). 

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]. 

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). ... 

By using the parabolic valley model, an expression of a (h v) for each type of transition (allowed or forbidden, depending on the selection rules) has been obtained [60-63]. The absorption coefficient, a h v), depends on the density of states for bands containing the initial and final states. The direct energy gap is a prominent feature of the experimental absorption data, due to its rapid variation with energy. The forbidden energy gap of the semiconductor is readily deduced... 

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... 

Hint is the optical interaction Hamiltonian given by Hi t=—er-E, where r is the position vector and E is the optical field. c) and v) represent the states in the conduction and valence bands, respectively, and i) is the virtual state within the transparency region. A simple model with a parabolic conduction band and a parabolic valence band gives for zinc blende semiconductors [226]... 

---