ELSEVIER Physica B 203 (1994) 201-213 PHYSICA Phase coherent transport in mesoscopic superconducting structures C.J. Lambert School of Physics and Materials, University of Lancaster, Lancaster, LA 1 4 YB, UK Abstract An overview of microscopic current-voltage relations applicable to mesoscopic superconductors is presented. These are used to examine a variety of new phenomena, including the change 8G in the two-probe electrical conductance G of a mesoscopic sample due to the switching on of superconductivity. It is predicted that 8G can have an arbitrary, sample dependent sign, have a magnitude much greater than 2e2/h and is suppressed by the application of a magnetic field. For an Andreev phase gradiometer formed by attaching a finite width normal wire at 90 ° to a superconductor, it is predicted that due to quantum interference from an order parameter phase gradient, the conductance of the wire will be an oscillatory function of the supercurrent. For an Andreev interferometer obtained by embedding a pair of superconductors with an order parameter phase difference $, in a disordered normal host, it is predicted that the phase periodic conductance G(~b) may have a maximum or a minimum at q~ = 0. In addition, the amplitude of the ensemble-averaged, 2rt periodic Fourier component decreases with energy, suggesting the possibility of a cross-over from a 2n to n periodicity with increasing temperature. Finally for a T-shaped normal structure, with a superconducting island located on the vertical leg and a current passing horizontally from left to right, it is predicted that the differential conductance exhibits a slow oscillatory dependence on the position of the superconductor and on the applied voltage. 1. Introduction The coming together of the hitherto separate fields of mesoscopic physics and superconductivityhas led to the dis- covery of many new phenomena. Some of these, such as non-local geometric oscillations [1,2], are generalisations of phenomena known to occur in macroscopic supercon- ductors [3, 4]. Others, such as anomalous proximity effects [5-11], zero bias anomalies [12-17], phase periodic trans- port [18-26] and quantized supercurrents [27-29] appear to be unique to mesoscopic structures. The aim of this pa- per is to highlight some recent theoretical results arising from the Lancaster/DRA collaboration [30] and to outline a conceptual framework for describing the above phenom- ena, based on microscopically derivable current-voltage re- lations [31 ]. These current-voltage relations are deceptively simple, since they do not address the problem of comput- ing the various quantum mechanical scattering coefficients contained therein. In a sense, they should be viewed as a convenient common ground, where theoreticians and exper- imentalists might meet to compare results. For experiments 0921-4526/94/$07.00 (~ 1994 Elsevier Science B.V. All rights reserved .~DlOOT'~-a~776(94~O0303-3 involving a single superconducting-normal(S-N) interface, they yield the well-known BTK formula for the electrical conductance ofa N-S interface [32]. For phase coherent N- S-N structures, they represent a generalization of two probe, Landauer-Bfittiker formulae [33]. In the presence of many normal probes [34], they yield a description of quasi-particle charge imbalance in phase coherent structures and for N-S- N-S-N structures, underpin a new description of the Joseph- son effect in weakly linked, mesoscopic, superconducting dots [35]. From a theoretical point of view, all scattering coefficients are derivable from the Bogoliubov~le Gennes equation or equivalently the Gorkov equations. In samples for which the order parameter A divided by the Fermi en- ergy EF is typically much less than unity, quasi-classical equations [36] might also be employed, although it should be noted that certain mesoscopic phenomena are not con- mined within such an approach. For example at a clean N-S interface, where quasi-classical theory allows only Andreev scattering, by solving the Bogoliubov~le Gennes equation exactly, it is readily shown that quasi-particles incident at low angles can be normally reflected with probability unity.