Electrical Conductivity of Solids

Abstract

The electrical and optical properties of a crystalline solid are intimately related to the energy-band characteristics of the substance. The conduction mechanism in metals and semiconductors is examined.

Questions

• 1. Give a description of the energy bands of metals, insulators and semiconductors. On the basis of the energy-band diagram, explain the electrical conductivity of each of these types of solid.

• 2. Explain why diamond is transparent while sodium and germanium are opaque. (Energies of photons in the visible spectrum: 1.5-3 eV.)

• 3. Consider three different crystals. Crystal A absorbs all electromagnetic radiation up to and including optical frequencies; crystal B absorbs radiation whose photons have energies of at least 5.9 eV; crystal C absorbs radiation with energies of at least 0.8 eV. Make a qualitative diagram of the upper energy bands for each crystal and describe the electrical and the optical properties of these crystals. (Energies of photons in the visible spectrum: 1.5-3 eV.)

• 4. Consider two crystals. Crystal A absorbs all electromagnetic radiation up to and including optical frequencies, while crystal B absorbs radiation whose photons have energies of at least 0.9 eV. The crystals are brought from a low-temperature region to a high-temperature region. What effect will this transfer have on the conductivity of each crystal?

• 5. Describe the physical significance of each of the following concepts:

  • (a) Conduction band of a metal.

  • (b) Valence and conduction bands of a semiconductor.

  • (c) Energy gap in a semiconductor.

  • (d) Hole in a semiconductor.

• 6. By using the general form of Ohm’s law, derive the familiar form, I=V/R, of this law for a metal wire of constant cross-section.

• 7. On the basis of Ohm’s law, explain why a metal is much more conductive than an intrinsic semiconductor at normal temperatures.

• 8. Describe the effect of temperature changes on the conductivity of metals and semiconductors. How do superconductors differ from ordinary metals in this respect?

• 9. Describe the physical mechanism by which an n-type or a p-type doping contributes to the conductivity of a semiconductor. What are the minority carriers in each case?

• 10. On the basis of the mass-action law, explain why by doping a pure semiconductor with impurities the conductivity of the substance is increased.

• 11. Consider a sample of unit volume of a crystal of pure germanium (Ge). The number of mobile electrons in the sample is equal to α. While keeping the temperature constant, we replace N atoms of Ge with phosphorus atoms (Ρ, 15) and another 10 Ν atoms of Ge with boron atoms (Β, 5). What will be the number of electrons after the doping process is completed?

• 12. As argued in Sect. 2.2, diamond is transparent since its energy gap E_G exceeds the energy of photons of visible light, making it impossible for such photons to be absorbed by the crystal. Instead of being absolutely transparent, however, some diamonds have a deep blue color due to the presence of boron atoms (Β, 5) in the crystal. In what way do the boron impurities alter the energy-band diagram of pure diamond? Make a qualitative diagram for the blue diamond, taking into account that E_G ≃ 6eV and that the photons of the “red” region of the optical spectrum have energies of the order of 1.7 eV. (Assume approximately that light consists of a “red” and a “blue” component.)

• 13. In what respect are diffusion currents different from currents obeying Ohm’s law? How are the signs in the expressions (2.24) for diffusion currents justified?