The change of resistance of a semi-conductor in a magnetic field

Abstract

One may say that prior to the introduction of the Fermi-Dirac statistics into the theory of metallic conduction and allied phenomena a general mathematical method of attack on the various problems had been developed which necessarily still forms the basis of the modern treatment; but nevertheless in most cases the older theory had little success in predicting the order of magnitude, and in some cases, even the qualitative features of the various effects. However, the ground had been well prepared, so that as soon as it was realized that the electrons in a metal did not really obey the Maxwell but the Fermi-Dirac statistics, the mere introduction of the latter distribution function in the place of the former in the classical equations proved sufficient to clear away many of the old difficulties. Since the appearance of Sommerfeld’s paper in 1928 the first order effects have received on the whole a satisfactory explanation. In the case of the second order effects, however— and it is with one of these that the present paper deals—there are still very considerable difficulties to be faced. The problem of the change of resistance of a metal in a magnetic field has been treated by Sommerfeld, making use of a method which was originally developed by Gans. The calculations follow closely the classical treatment of Lorentz in that the mean free path of an electron is introduced phenomenologically as a parameter to be determined from the known experimental value of the conductivity. In the classical theory one pictures the process as follows. The metal is regarded as having a regular three-dimensional lattice structure with the metallic ions situated at the lattice points. It is further supposed that there are a certain number of conduction electrons, which might well correspond with the valency electrons, and that the assembly of conduction electrons obeys the classical distribution law. When an electric field is applied in a given direction the electrons are accelerated and experience elastic collisions with the metallic ions. Finally an equilibrium state is reached in which the number of electrons entering a given velocity range in unit time is just equal to the number ejected by collisions, and the mathematical expression of this state takes the form of an integral equation which must be solved to find the change in the original distribution function due to the applied field. From the change in the distribution function the conductivity is calculated. In the semi-classical calculations of Sommerfeld the model is the same except that the Fermi-Dirac statistics are used instead of the Max-wellian. If one compares the value of the conductivity, thus obtained, with the experimental value, one obtains a mean free path which is about a hundred times greater than the lattice spacing. This large value is not very plausible on classical ideas; but is readily understandable on wave mechanical principles.