An isolated atom of germanium has 32 electrons, arranged in subshells according to this scheme:$\mathbf{1}{\mathit{s}}^{\mathbf{2}}\mathbf{2}{\mathit{s}}^{\mathbf{2}}\mathbf{2}{\mathit{p}}^{\mathbf{6}}\mathbf{3}{\mathit{s}}^{\mathbf{2}}\mathbf{3}{\mathit{p}}^{\mathbf{6}}\mathbf{3}{\mathit{d}}^{\mathbf{10}}\mathbf{4}{\mathit{s}}^{\mathbf{2}}\mathbf{4}{\mathit{p}}^{\mathbf{2}}$ This element has the same crystal structure as silicon and, like silicon, is a semiconductor. Which of these electrons form the valence band of crystalline germanium?

Show that the probability P(E) that an energy level having energy E is not occupied is$\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{e}}^{\mathbf{-}\mathbf{∆}\mathbf{EIkT}}\mathbf{}\mathbf{+}\mathbf{1}}\mathbf{where}\mathbf{}\mathbf{∆}\mathbf{E}\mathbf{=}\mathbf{E}\mathbf{-}{\mathbf{E}}_{\mathbf{F}}$ where .

What is the probability that, at a temperature of T = 300 K, an electron will jump across the energy gap ${\mathbf{E}}_{\mathbf{g}}\left(=5.5\mathrm{eV}\right)$ in a diamond that has a mass equal to the mass of Earth? Use the molar mass of carbon in Appendix F; assume that in diamond there is one valence electron per carbon atom.

Doping changes the Fermi energy of a semiconductor. Consider silicon, with a gap of 1.11eV between the top of the valence band and the bottom of the conduction band. At 300K the Fermi level of the pure material is nearly at the mid-point of the gap. Suppose that silicon is doped with donor atoms, each of which has a state 0.15eV below the bottom of the silicon conduction band, and suppose further that doping raises the Fermi level to 0.11eV below the bottom of that band (Fig. 41-22). For (a) pure and (b) doped silicon, calculate the probability that a state at the bottom of the silicon conduction band is occupied. (c) Calculate the probability that a state in the doped material (at the donor level) is occupied.

What is the number density of conduction electrons in gold, which is a monovalent metal? Use the molar mass and density provided in Appendix F.

Verify the numerical factor 0.121 in Eq. 41-9.