Dephasing of Cooper pairs and subgap electron transport in superconducting hybrids

Abstract

We argue that electron-electron interactions fundamentally restrict the penetration length of Cooper pairs into a diffusive normal metal (N) from a superconductor (S). At low temperatures, this Cooper pair dephasing length L ϕ remains finite and does not diverge at T → 0. We evaluate the subgap conductance of NS hybrids in the presence of electron-electron interactions and demonstrate that this length L ϕ can be directly extracted from conductance measurements in such structures. It is well known that a normal metal (N) attached to a superconductor (S) also acquires superconducting properties. 1,2This superconducting proximity effect is directly related to the phenomenon of Andreev reflection:3 At the NS interface Cooper pairs are converted into subgap quasiparticles (electrons), which can diffuse deep into the normal metal keeping information about a macroscopic phase of the superconducting condensate. Such macroscopic quantum coherence of electrons in the normal metal gets destroyed by thermal fluctuations, provided the corresponding inverse electron diffusion time (Thouless energy) becomes smaller than temperature T . As a result, superconducting coherence extends into a normal metal at a typical length L T ∼ √ D/T (where D is the electron diffusion coefficient), implying that the whole normal metal can demonstrate superconducting properties at sufficiently low T .This proximity-induced superconductivity manifests itself in a number of well-known phenomena, such as Meissner and Josephson effects in normal-superconducting hybrids 4,5 as well as dissipative transport of subgap electrons across NS interfaces.6 Provided the NS interface transmission is low, its corresponding subgap (Andreev) conductance G remains rather small, being proportional to the second order in the barrier transmission. On the other hand, G can be strongly enhanced at low energies due to nontrivial interplay between disorder and quantum interference of electrons in the normal metal, 7-11 which leads to the so-called zero-bias anomaly (ZBA) G ∝ 1/ √ V and G ∝ 1/ √ T in the limit of low voltages and temperatures.In this paper we will demonstrate that in the lowtemperature limit both superconducting proximity effect and ZBA in Andreev conductance are limited by dephasing of Cooper pairs due to electron-electron interactions in the normal metal. Note that Coulomb effects in subgap electron transport across NS interfaces have been studied in several works;11-14 however, the decoherence effect of Coulomb interaction has not yet been addressed in a proper and complete manner. Below we will argue that fluctuating electromagnetic field produced by fluctuating electrons in a disordered normal metal destroys macroscopic coherence of electrons penetrating from a superconductor at a typical length scale L ϕ . The existence of this length scale imposes fundamental limitations on the proximity effect in NS hybrids at low temperatures T D/L 2 ϕ . In this temperature range, the penetration depth of superconducting correlations into the normal metal is not ...

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