On the thermo-electric power and thermal conductivity of semi-conductivity of semi-conductors and metals

Abstract

The first successful attempt to explain the electrical and thermal properties of metals was made by the Drude-Lorentz theory at the beginning of this century. According to Drude, certain electrons were free to move from atom to atom throughout the metal and it was those electrons which undertook the conduction of electricity and heat. The electrons were then treated as a “gas” and in order to apply the statistical theory of gases to such an electron cloud Lorentz postulated his well-known assumptions. The theory was immediately successful in the derivation of the Wiedemann-Franz law, relating the electrical and thermal conductivities. The major drawback to the theory lay in the evaluation of the electron specific heat. Lorentz had ascribed to the electrons a Maxwell distribution of velocities, the only reasonable choice at that time. On such a picture the electron specific heat was large and such an addition to the specific heat completely destroyed the agreement of Debye’s theory with the experimentally observed specific heats.

The theory remained in this state until the discovery by Pauli, in 1925, of the Pauli Exclusion Principle. In its simplest form the principle states that in an atom not more than two electrons can have the same three quantum numbers. This allowed Dirac and Fermi, working independently, to develop the statistics of particles obeying such a principle and gave birth to the Fermi-Dirac statistics. The use of new statistics enabled the discrepancy in the specific heats to be explained. Pauli was able to account for the paramagnetism of the alkali metals and it was left to Sommerfeld to consider the problems of transport phenomena in the light of the Fermi-Dirac statistics.

Such were the foundations of the modern electron theory of metals. The modern development of the theory was begun by Block, Sommerfeld, Bether, Peierls and Wilson. It is based on the quantum-mechanical analysis of the motion of an electron in the periodic field of a crystal lattice. Considered from this point of view, the electrons in a metal are distributed over a number of allowed energy bands, forbidden bands occurring in the regions between the allowed energies. Most of these allowed bands are filled completely by the electrons, and it is only the electrons which are contained in incompletely filled bands which contribute to the resultant current. It is these electrons which are regarded as “free” in conduction theory. This picture of a metal also enables us to obtain a better understanding of the “mean free path” of an electron, a quantity which is treated as an arbitrary parameter in the Sommerfeld theory.