On the relation between direct and inverse methods in statistics

Abstract

The work of Pippard, Shepherd & Tindall (P.S.T.) on the resistance of copper foil sandwiched between superconducting lead electrodes has been extended (1) to study the influence of alloying the lead with bismuth, (2) to investigate other pairs of metals, Pb-Re, Pb-Mo, Pb-Al, Pb-Sn, Sn-Al. In addition, different techniques for depositing lead electrodes on copper foil have been investigated; evaporation in vacuo after cleaning the copper surface by sputtering has been found to give the most consistent results and minimal oxide contamination at the metallic interface. Thin oxide layers give rise to a characteristic rise in the resistance of the sandwich as the temperature is lowered well below the superconducting transition. The behaviour is fairly consistent with a simple model in which regions of metallic contact carry current in parallel with tunnel junctions; the conductivity of the latter is much higher than for oxide layers used for experiments on superconducting tunnelling, and it is conjectured that they are monomolecular. The rise of resistance towards the transition temperature of lead, as observed by P.S.T., becomes more marked when bismuth is added to the lead, but reaches a limiting behaviour with 5% Bi which is maintained up to 20% Bi, at which point the behaviour becomes less reproducible, as might be expected with a two phase superconducting electrode. The resistance of the sandwich at the lowest temperatures is also somewhat increased by the addition of bismuth to the lead. A theory of the observations has been developed by extension of the argument of P.S.T. which attributes the extra resistance to the penetration of excitations from the normal metal into the superconductor, provided they have an energy above the gap. With the addition of bismuth these excitations are more frequently scattered before finally being assimilated into the equilibrium superconducting state, and the extra scattering reveals itself as extra resistance. At the lowest temperatures no excitations can enter the superconductor from the normal metal, and if the superconductor were pure and metallic contact perfect the total reflexion of excitations with reversal of character, as analysed by Andreev, would lead to no extra resistance. The reflected excitations, however, penetrate to something like the coherence length into the superconductor in the form of an evanescent mode. It is the scattering of the evanescent mode by impurities which is responsible for partial reflexion without change of character, and hence for the extra resistance at low temperatures. The scattering process for evanescent modes is analysed in some detail and the problem of developing a full theory in the three dimensional case is discussed but without a final conclusion. The one dimensional model is nevertheless capable of fairly complete solution, though reasons are given for not trusting its predictions in the real situation. Although fully quantitative explanation of the experimental results is not possible either at low temperatures or near the transition temperature (in the latter case because of effects due to short lifetime of excitations in lead), the qualitative agreement between models and experiment is not unsatisfactory. The results on sandwiches in which the centre part becomes superconducting at a lower temperature than the flanking electrodes confirm other observations in showing how the resistance begins to drop as the temperature falls towards the lower transition temperature. The Pb-Mo samples behave as might have been expected if the molybdenum had a transition temperature of about 1 K although these samples, possibly due to iron content, had a transition temperature below 0.4 K.