3 - Electrons in a metal at finite temperatures

Summary

Prologue

In Chapter 2, we constructed the Fermi sphere of free-electrons with the radius k_F in reciprocal space. It represents the distribution of the quantized electronic states at absolute zero, in which the states in k≤k_F are all occupied but those in k>k_F are vacant. At finite temperatures, thermal energy would excite some electrons in the range k≤k_F into the range k>k_F. The redistribution of electrons will occur so as not to violate the Pauli exclusion principle. As noted in Section 2.7, the Fermi energy in typical metals is of the order of several eV and is equivalent to ∼ 10000K on the temperature scale. Hence, only electrons near the Fermi surface can be excited at temperatures below ∼ 1000K. The aim of the present chapter is to formulate first the Fermi–Dirac distribution function, which determines the distribution of electronic states or the Fermi surface at finite temperatures, and then to deduce the temperature dependence of various physical properties due to conduction electrons by calculating relevant quantities involving the Fermi–Dirac distribution function.

Fermi–Dirac distribution function (I)

We know that the velocity of dilute gas molecules obeys the Maxwell–Boltzmann distribution law. Unfortunately, however, classical statistics cannot be applied to the conduction electron system in metals because of an extremely high electron density of the order of 10²⁸–10²⁹/m³. As emphasized in the preceding chapter, an electron carries a spin of ½ and particles with a half-integer spin should obey the Pauli exclusion principle. In addition, they are indistinguishable from each other. Our first objective in this section is to deduce the statistical distribution function under these two conditions imposed by quantum mechanics.