The Bandgap Energy

Once the sp hybrid is formed, all the electrons are tied up as valence electrons forming saturated covalent bonds. Since these tend to be strong bonds, especially in the case of C, there is a significant bandgap between the valence band and the conduction band because one or more of the covalent bonds must be broken to provide a conduction electron. Thus, 

Since the bandgap energy increases as the lattice parameter decreases, one might expect that bandgap energy would increase with pressure. Conversely, since the lattice parameter increases with temperature due to thermal expansion, one would expect the bandgap energy to decrease with temperature. 

Cohesive Energies and Bandgap Energies for Group IV Semiconductors 

Source Data from Sapoval, B. and Hermann, C., Physics of Semiconductors, Springer-Verlag, New York, 1995. 

---

The IR band has been ascribed to radiative recombination from the CB to a DB center [Ku6]. Such a recombination process is sensitive to changes in the CB energy only and is therefore expected to show a weaker dependence on confinement than the red PL. This interpretation, together with the observation that the increase in VB energy is about twice the increase in CB energy, as shown in Fig. 7.16, has been used to approximate the bandgap energy in micro PS, as shown in Fig. 7.13. 

A first test of this model was performed with pressure experiments on InP Yb3+. Here the Yb3+ ion introduces an electron trap to the semiconductor host. The pressure-induced shift of the 2F5/2 -> 2Ft/2 intra 4f shell transitions amounts to 0.96 meV/GPa up to 4 GPa (Stapor et al., 1991), while the bandgap energy of InP increases by 85 meV/GPa (Trommer et al., 1980). 

Fig. 1. (a) Diffuse reflectance spectra of P25 (thin line), TH (thick line), 3% [PtClJ/P25 (dashed line) and 4.0% H2[PtCl6]/TH (dotted line). The Kubelka-Munk function, F(R00), is used as the equivalent of absorbance, (b) Transformed diffuse reflectance spectra of P25 (thin line), TH (thick line), 3% [PtCl4]/P25 (dashed line) and 4.0% H2[PtCl6]/TH (dotted line). The bandgap energy was obtained by extrapolation of the linear part. 

Absorption of a photon with an energy hv greater or equal the bandgap energy Eg of the semiconductor (i.e., 3.2 eV for titanium dioxide in its anatase modification) generally leads to the formation of an electron/hole pair in the semiconductor particle (reaction (7.1) and Fig. 7.1). 

The primary events occurring within a nanometer-sized semiconductor particle after the absorption of a photon the energy of which is exceeding the bandgap energy have been discussed in detail based upon a review of the current literature. Both, electrons and holes, are separated extremely rapidly from the initially formed exciton and trapped at or very close to... 

Unlike atomic or solid-state lasers, the lasing transitions in a semiconductor laser are transitions between continua of extended states rather than between localised states. The inversion criterion [4] then is that the electron and hole quasi-Fermi levels must be separated by more than the bandgap energies. The spectrum of the optical gain g is given by [5,6]... 

The n parameter equals 1 for direct bandgap semiconductors or 4 for indirect bandgap semiconductors in the case of allowed fundamental transitions [22], Other values of n, 2 or 3, are valid only for forbidden transitions. The proper transformation allows estimation of the bandgap energy, Eg, for both types of crystalline semiconductors. Figure 7.7 presents the procedure of Eg evaluation. 

Further complication in semiconductor band shape analysis concerns the spectral region near the fundemental absorption onset. Ideal semiconductor crystal at 0 K should not absorb any photons with energies lower than Eg. Real systems, however, show pronounced absorption tails at energies lower than the bandgap energy (Figure 7.7). The absorption profile within the tail region can be very well approximated by the empirical Urbach s rule [23-26] ... 

Figure 7.7 (a) Absorption spectrum of a semiconductor and (b) its transformation to (aE)2,n vs E. Ehe red line shows extrapolation of the linear part of the absorption band edge. Its crossing with the energy axis determines the bandgap energy Eg... 

Figure 7.13 Energy diagrams for donor-acceptor surface-molecule interactions in the case of (a) n-type semiconductor-molecular donor and (b) p-type semiconductor and molecular acceptor. Eg denotes the bandgap energy and EF the Fermi energy. See text for other details...

The semiconductor electrode most studied in photoelectrolytic cells has been n-Ti02, and in photogalvanic cells n-Sn02. Because the bandgap energies are 3.0 eV and 3.5 eV respectively, they are not optimum semiconductors as they only make use of about 5 per cent of the solar energy. For this reason there has been research into other semiconductors, for example cadmium sulphide. In all cases the efficiency is fairly low. 

---