Anisotropy of Relaxation Times and Phonon Drag in the Noble Metals

The effect of impurities on the phonon-drag part $S_g$ of the thermoelectric power at very low temperatures is examined for alloys of copper, silver, and gold. $S_g$ is separated into three terms corresponding to: (1) the neck regions in the $〈111〉$ directions, (2) the convex belly regions around the $〈100〉$ directions, and (3) the concave belly regions around the $〈110〉$ directions. We find that in gold the concave belly region is most important and in silver the necks are most important. The sign of the neck contribution is examined through the behavior of the $〈110〉$ phonons at the zone boundary, and it is found that the positive contributions dominate there. The experimental results for three types of impurities, (A) uncharged, (B) charged, and (C) transition-metal, are discussed, and qualitative agreement with the theory is found.