Reentrance of the Metallic Conductance in a Mesoscopic Proximity Superconductor

Abstract

We present an experimental study of the diffusive transport in a normal metal near a superconducting interface, showing the reentrance of the metallic conductance at very low temperature. This new mesoscopic regime comes in when the thermal coherence length of the electron pairs exceeds the sample size. The reentrance is suppressed by a bias voltage given by the Thouless energy and can be strongly enhanced by an Aharonov-Bohm flux. Experimental results are well described by the linearized quasiclassical theory. [S0031-9007(96)01817-0] PACS numbers: 74.50.+r, 73.20.Fz, 73.50.Jt, 74.80.Fp During the past few years, the proximity effect between a superconductor (S) and a normal (N) metal has met a noticeable revival, thanks to spectacular progress in the fabrication of samples of mesoscopic size [1]. Experimental study of the transport near a S-N interface has shown that the proximity effect strongly affects electron transport in mesoscopic S-N systems: The deviation DG of the conductance from its normal-state value depends strongly on temperature T and oscillates in an applied magnetic field H if a N loop is present [2][3][4]. Various theoretical approaches were suggested to explain this behavior. A scattering matrix method based upon the Landauer formula [5] as well as a numerical solution of the Bogolubov-de Gennes equations [6] were used. These studies demonstrated that superconductivity does not affect the charge transfer in the N metal if the temperature T and the voltage V are zero, i.e., DG is zero at zero energy. A more powerful method based on the equations for the quasiclassical Green's functions [7-10] was used to obtain the dependence of DG on T and V. It has been established [9] that at V 0 the deviation of the conductance DG increases from zero at T 0 (if electron-electron interaction in N is negligible) with increasing T, reaches a maximum at approximately the Thouless temperature e c ͞k B hD͞k B L 2 , and decreases to zero at T ¿ e c ͞k B . This constitutes the reentrance effect for the metallic conductance of the N metal. Similar dependence of DG͑V ͒ at T 0 has been found in [10] both in a numerical solution of the Bogoliubov-de Gennes equations and in an analytical solution of the equations for the quasiclassical Green's functions.The physics behind this reentrance effect involves nonequilibrium effects between quasiparticles injected by the N reservoirs and electron pairs leaking from S. At the N-S interface, and incident electron is reflected into a hole of the same energy e compared to the Fermi level E F , but with a slight change in wave vector dk due to the branch crossing: dk͞k F e͞E F , k F being the Fermi wave vector. The phase conjugation between the electron and the hole results in a finite pair amplitude involving states ͑k F 1 e͞hy F , 2k F 1 e͞hy F ͒, y F being the Fermi velocity. Such a pair maintains coherence in N up to the energy-dependent diffusion length L e ph D͞e [7,11] which coincides with the well-known thermal length L T ph D͞k B T at e 2pk B T . In the high-temperature...

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