The influence of small-angle scattering on metallic conduction

Abstract

Certain transport effects are known to be sensitive to the type of scattering responsible for resistance, so that the relaxation-time approximation becomes inadequate. It is argued that small-angle scattering, such as is caused by low-temperature phonons and extended defects, may be as effective as isotropic scattering if the disturbance of the distribution function by the applied field varies abruptly over the Fermi surface. Any scattering process between points differing greatly in their disturbed distribution, however close together in k -space, creates entropy and is an effective collision. A number of examples are discussed in qualitative terms from this viewpoint, especially the size-effect and various magnetoresistance phenomena in which the Fermi surface topology and the magnetic field together create the required conditions for abrupt variations. In the second section, several special models are analysed in detail: (1) The free-electron gas acted on by a spatially periodic electric field; the conductivity is found to differ by at most 20% between conditions of isotropic and small-angle scattering, in spite of considerable effects of the scattering function on the form of the disturbed distribution. (2) Transverse magnetoresistance in a model metal whose Fermi surface resembles that of copper, etc., in generating alternate belts of electron and hole orbits. Scattering between the belts delays saturation of the resistance, which in very strong fields may rise to a much higher value than for isotropic scattering. If open orbits are present, however, the non-saturating quadratic increase with field is diminished by small-angle scattering. (3) Transverse magnetoresistance in a model metal with a nearly free electron Fermi surface just overlapping the first Brillouin zone (e. g. Al, Mg, Zn). Small-angle scattering between zones produces effects similar to those in (2), though the analysis is quite different. In the third section, some of the scanty published data on Cu, In and Al are discussed briefly to indicate that the puzzles they present may be resolved if small-angle scattering (or equivalent effects due to mosaic structure) dominates the behaviour of ultra-pure samples. But there is need for more experiment before anything can be said for certain.