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Coherent transpory in a normal wire between reservoirs

F.K. Wilhelm, Andrei D. Zaikin, A.A. Golubov

1996

We develop a detailed analysis of electron transport in normal di usive conductors in the presence of proximity induced superconductivity. A rich structure of temperature and energy dependencies for the system conductance, density o f states and related quantities was found and explained. If the normal conductor forms a loop its conductance changes h=2e-periodically with the magnetic ux inside the loop. The amplitude of these conductance oscillations shows a reentrant behavior and decays as 1=T
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... at high T . Presently the transport properties of normal superconducting proximity systems attract much experimental and theoretical interest. Various nontrivial features of such systems have been recently discovered 1, 2 The aim of this paper is to investigate the coherent electron transport in a normal di usive conductor attached to a superconductor. One can show 3 that complicated geometrical realizations of the system 1, 2 can be essentially reduced to the following simple model: a normal diffusive wire is attached to a normal reservoir at x = 0 and a superconducing one at x = d. In order to calculate the conductance of this wire we use the standard formalism of quasiclassical Green functions in the Keldysh technique see e.g. 4 . The rst step is to nd the retarded normal and anomalous Green functions of the system g R = cosh and f R = sinh , = 1 + i 2 . In the di usive approximation this has been done with the aid of the Usadel equation see 3 for details. The second step is to solve the kinetic equation. As a result for a di erential conductance of the system normalized to its Drude value G N at low v oltages and in the absence of tunnel barriers one nds 4 G = 1 2T R 1 1 is the transparency of the system at the energy . The conductance GT shows the reentrant b ehavior Fig. 1 see also 5 . At l o w temperatures T d the correction is G := G , 1 T= d 2 , at T d we h a v e G p d =T . The square-rootscaling of G at high T has an obvious physical interpretation: as the part of the N-wire of the size N = p D=2T close to the NS boundary becomes e ectively superconducting due to the prox-0.000 0.002 0.004 0.006 0.008 0.010 ε/∆ (ε d =10 -4 ) 0.0 0.2 0.4 0.6 0.8 1.0 N/N(0) Normal Correlation 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 T/∆ 0.00 0.02 0.04 0.06 0.08 0.10 Fig.2 imity e ect, only the rest of the wire contributes to the resistance. As a result it becomes smaller than 1=G N . As the temperature is lowered the conductance increases, reaches its maximum at N d an then decreases again. In order to understand the reentrant behavior of G at low T d w e calculated the density o f states DOS averaged over the length of the wire. The normal density of states N N = N 0 R 1 0 d x g shows a soft pseudogap below d see Fig. 2 . At the rst sight a t l o w T this would lead to a decrease of G below 1. This is, however, not the case because of an additional contribution of correlated electrons present in the N-wire due to the proximity e ect. The DOS for such electrons in the wire N S = N 0 R 1 0 d x =f becomes larger for small Fig.2. These two e ects exactly compensate each other at T = 0, in which case G = 1 . F or T 0 w e always have g 2 + = f 2 = cosh 2 1 1, i.e. the pseudogap e ect never dominates the correlationinduced enhancement and GT 0 1. On the other hand, due to the presence of this pseudogap at d the total transparency D decreases with

doi:10.5445/ir/180496
fatcat:ochf5qmymjdi5jkjpvmvizzrvy