A semiconductor has an electrical resistivity that is intermediate between those of good conductors and of good insulators. The tremendous importance of semiconductors in present-day electronics stems in part from the fact that their electrical properties are very sensitive to very small concentrations of impurities. We’ll discuss the basic concepts using the semiconductor elements silicon (Si) and germanium (Ge) as examples.Silicon and germanium are in Grour IV of the periodic table. Each has four electrons in the outermost electron subshells (3s3l for Si, 4i4l for Ge). Both crystallize in the diamond structure (Section 44-4) with covalent bonding; each atom lies at the center of a regular tetrahedron, forming a covalent bond with each of four nearest neighbors at the comers of the tetrahedron. Because all the valence electrons are involved in the bonding, at absolute zero the band structure (Section 44-5) has a completely filled valence band separated by an energy gap from an empty conduction band (Fig. 44-lSc). This distribution makes these materials insulators at very low temperatures; their electrons have no nearby states available into which they can move in response to an applied electric field. However, the energy gap E. between the valence and conduction bands is small in comparison to the gap of 5 eV or more for many insulators; room temperature values are 1.42 eV for gallium arsenide, 1.12 eV for silicon, and only 0.67 eV for germanium. Thus even at room temperature a substantial number of electrons can gain enough energy to jump the gap to the conduction band, where they are dissociated from their parent atoms and are free to move about the crystal. The number of these electrons increases rapidly with temperature.
In principle, we could continue the calculation in Example 44-10 to find the actual density n = NIV of electrons in the conduction band at any temperature. To do this, we would have to evaluate the integral Jg(E}f(E) dE from the bottom of the conduction band to its top. First we would need to know the density of states function g(E). It wouldn’t be correct to use Eq. (44-16) because the energy-level structure and the density of states for real solids are more complex than those for the simple free-electron model. However, there are theoretical methods for predicting what g(E) should be near the bottom of the conduction band, and such calculations have been carried out. Once we know n, we can begin to determine the resistivity of the material (and its temperature dependence) using the analysis of Section 26-3, which you may want to review. But next we’ll see that the electrons in the conduction band don’t tell the whole story about conduction in semiconductors.