In a real semiconductor, the density of states at the bottom of the conduction band will differ from the model used in the previous problem by a numerical factor, which can be small or large depending on the material. Let us therefore write for the conduction band g(e) = 90CVE - €c, where gọc is a new normalization constant that differs from go by some fudge factor. Similarly, write g(e) at the top of the valence band in terms of a new normalization constant gov. (a) Explain why, if gov # 90c, the chemical potential will now vary with tem- perature. When will it increase, and when will it decrease? (b) Write down an expression for the number of conduction electrons, in terms of T, µ, Ec, and goe. Simplify this expression as much as possible, assuming Ec - u > kT. (c) An empty state in the valence band is called a hole. In analogy to part (b), write down an expression for the number of holes, and simplify it in the limit u - Ev » kT. (d) Combine the results of parts (b) and (c) to find an expression for the chemical potential as a function of temperature. (e) For silicon, gọc/go = 1.09 and gov/g0 = 0.44.* Calculate the shift in u for silicon at room temperature.

In a real semiconductor, the density of states at the bottom of the conduction band will differ from the model used in the previous problem by a numerical factor, which can be small or large depending on the material. Let us therefore write for the conduction band g(e) = 90CVE - €c, where gọc is a new normalization constant that differs from go by some fudge factor. Similarly, write g(e) at the top of the valence band in terms of a new normalization constant gov. (a) Explain why, if gov # 90c, the chemical potential will now vary with tem- perature. When will it increase, and when will it decrease? (b) Write down an expression for the number of conduction electrons, in terms of T, µ, Ec, and goe. Simplify this expression as much as possible, assuming Ec - u > kT. (c) An empty state in the valence band is called a hole. In analogy to part (b), write down an expression for the number of holes, and simplify it in the limit u - Ev » kT. (d) Combine the results of parts (b) and (c) to find an expression for the chemical potential as a function of temperature. (e) For silicon, gọc/go = 1.09 and gov/g0 = 0.44.* Calculate the shift in u for silicon at room temperature.