In a real semiconductor; the density of states at the

Chapter 7, Problem 34P

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Problem 34P

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  g0c is a new normalization constant that differs from go by some fudge factor. Similarly, write g(ϵ) at the top of the valence band in terms of a new normalization constant gov.

(a) Explain why, if g0u ≠ I g0c, the chemical potential will now vary with temperature. When will it increase, and when will it decrease?

(b) Write down au expression for the number of conduction electrons, in terms of T, µ ϵc, and g0c Simplify this expression as much as possible, assuming ϵc − µ ≫ 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 µ − ϵv ≫ KT.

(d) Combine the results of parts (b) and (c) to find an expression for the chemical potential as a function or temperature.

(e) For silicon g0c/g0 = 1.09 and g0v/g0 = 0.44* Calculate the shift in µ for silicon at room temperature.