Electron-hole imbalance in superconductor/normal-metal mesoscopic structures

Abstract

We analysed the electron-hole or, in another words, branch imbalance (BI) and the related electric potential V imb which may arise in a mesoscopic superconductor/normal metal (S/N) structure under non-equilibrium conditions in the presence of a supercurrent. Non-equilibrium conditions can be created in different ways: a) a quasiparticle current flowing between the N reservoirs; b) a temperature gradient between the N reservoirs and no quasiparticle current. It is shown that the voltage V imb oscillates with the phase difference ϕ. In a cross-geometry structure the voltage V imb arises in the vertical branch and affects the conditions for a transition into the π−state.A few decades ago a great deal of interest was paid to the study of effects related to the so called branch imbalance (BI) (see Refs. [1,2] and for example the reviews [3,4]). The BI implies that populations of the electron-like and hole-like branches of the excitation spectrum in a superconductor or a normal metal are different. For example, the BI may arise in a superconductor near the S/N interface if a current flows through this interface and the temperature is close to T c . The conversion of the quasiparticle current j Q into the condensate current j S occurs over a rather long length λ Q called a BI relaxation length. Over this length populations of the electron-like and hole-like branches of the energy spectrum differ from each other. The difference between these populations is characterized by the distribution function f − = −(n ↑ − p ↓ ); the function n ↑ is the distribution function of the electron-like excitations and p ↓ (ǫ) = 1−n ↓ (−ǫ) is the distribution function of the hole-like excitations. In the considered case of a spin-independent interaction one has n ↑ = n ↓ and p ↑ = p ↓ . One can show that the function f − differs from zero if divj Q = 0. The non-zero distribution function f − leads to the appearance of an electric potential V imb in a superconductor (or in a normal metal) which can be expressed in terms of the function f − (see below). The BI may also arise in a bulk superconductor. For example, if longitudinal collective oscillations with a finite wave vector q are excited in the superconductor, the BI arises because in this case divj Q = iqj Q = 0 . When these modes are excited (they are weakly damped only near T c ), the quasiparticle current j Q oscillates in a counter phase with the condensate current j S , so that the total current remains equal to zero. These oscillations have been observed experimentally by Carlson and Goldman (Carlson-Goldman mode) [5] and have been explained theoretically in Refs. [6,7]. Another example of a system, in which the BI arises, is a uniform superconducting film in the presence of a temperature gradient ∇T and a condensate flow. It was established experimentally [8] and theoretically [9,10,11] that in this case the BI has a magnitude which is proportional to v s ∇T , where v s is the condensate velocity.Recently there has been growing interest in the study of transport properties of S/...

Peltier effects in Andreev interferometers

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