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bandgap materials are ideally suited for use in light-emitting diodes, semiconductor lasers, and other optical devices. The E versus k diagram for silicon is shown in Figure 2.18b. The maximum in the valence-band energy occurs at k=0, as before. The minimum in the conduction-band energy occurs not at k=0, but along the [100] direction. The difference between the minimum conduction-band energy and the maximum valence-band energy is still defined as the bandgap energy Eg. A semiconductor whose maximum valence-band energy and minimum conduction-band energy do not occut at the same k value is called and an indirect bandgap semiconductor. When electrons make a transition between the conduction and valence bands, we must invoke the law of conservation of momentum. A transition in an indirect bandgap material must necessarily include an interaction with the crystal so that crystal momentum is conserved. Germanium is also an indirect bandgap material, whose valence-band maximum occurs at k=0 and whose conduction-band minimum occurs along the [111] direction. GaAs is a direct bandgap semiconductor, but other compound semiconductors, such as Gap and ALAs, have indirect bandgaps. We can relate the simple line representation of the energy-band diagram shown in Figure 2.10b to the energy bands given in Figure 2.18. In the line representation of the energy-band diagram, the energies Ev and Ec correspond to the maximum valence-band energy and minimum conduction-band energy, respectively. These sane energies are shown in Figure 2.18. The line representation of the energy band does not show the direct or indirect bandgap nature of the semiconductor. 2.4 | DENSITY OF STATES FUNCTION Objective: Discuss and develop the density of quantum states as a function of electron energy. As we have stated, e eventually wish to describe the current-voltage characteristics of semiconductor devices. Since current is the result of the flow of charge, an important step in the proce