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An elementary approach to Fermi-Dirac statistics

Fermi-Dirac statistics is most commonly encountered in connection with electrons in metals and semiconductors. Many students have to learn about these materials without having had a previous grounding in statistical mechanics, and many elementary textbooks either quote the required results without proof or else attempt  some  form of simple and often indirect derivation. This article presents a way of approaching the subject which emphasises the essential statistical mechanical ideas but avoids a derivation of the results in mathematical terms. The  author  has used this approach in teaching a first-year class of electrical engineering undergraduates about semiconductors, and also in an introductory course on metals for physics students. The treatment is made possible by concentrating solely on the problem of Fermi-Dirac statistics. There are, of course, other types of statistics which apply to problems different from those discussed here, and in view of this restiction to one special field, the approach is not suggested as a way of introducing a course on statistical mechanics. It might, however, be useful in discussing the topic during such a course. In a single atom the extra-nuclear electrons can only occupy certain specific energy levels, and these levels are occupied according to the Pauli exclusion principle which states that there cannot be more than one electron in any one level. (We will count two quantum states with equal energy but  opposite electron spin as two different levels, and ignore the problem of electron spin entirely.) In the ground state of the atom the electrons occupy those levels which make the total energy a minimum; that is, if there are N electrons, they occupy the N levels with lowest energy. In metals and semiconductors the electrons cannot all be considered as associated withspecific atoms. A piece of the material must be considered as an entity, and the available electrons occupy the energy levels of this whole system. At  absolute zero the system is in its ground state with the N electrons occupying the N levels of lowest energy, but at higher temperatures some electrons occupy levels with higher energy, the additional energy required being part of the thermal energy of the material. It is the purpose of Fermi-Dirac statistics to predict how the available electrons will be distributed over a particular set of energy levels at any temperature. The pattern of energy levels in a metal or semiconductor is an important subject which is described in most books on solid state physics. This article will be concerned only with the statistical problem of how the levels are occupied. An essential feature of the problem is that the occupation of the levels is governed by the Pauli exclusion principle. For particles (such as  atoms) to which the exclusion principle does not apply the statistical problem is very different; it is known as Bose-Einstein statistics.