Analytical Derivation of the Excess Current in Fully Incoherent Josephson Junctions

In this work we present an analytical derivation of the excess current in Josephson microbridges, following the Octavio-Blonder-Tinkham-Klapwijk (OBTK) model. We can confirm previously found numerical results, including the prediction of negative excess current for low transparencies, which might possibly shed new light on questions concerning the validity of the OBTK approach. Introduction There are currently two main approaches to describe the current, excess current, subgap structure and other features of Josephson junctions theoretically. The first one is the so-called Octavio-BlonderTinkham-Klapwijk (OBTK) model, as outlined in [Blonder et al., 1982], [Octavio et al., 1983] and [Flensberg et al., 1988], which is mainly used for microbridges and similar systems. One of its characteristics is that all information about the evolution of the complex phase factor inside the junction area is lost. We therefore speak of fully incoherent Josephson junctions, even though the processes at each of the interfaces conserve phase. The other approach is a Hamiltonian model, as presented in [Cuevas et. al., 1996] and [Levy Yeyati et al., 1997], which is especially adapted to small systems like quantum point contacts and quantum dots, since it conserves phase entirely throughout the junction. In this present work we will follow the OBTK model and reproduce the excess current found by [Flensberg et al., 1988] in a different way. We briefly discuss the validity of these findings and their apparent incongruity with results obtained using the Hamiltonian approach. The OBTK model Our calculations are based on the work done by [Blonder et al., 1982], [Octavio et al., 1983] and [Flensberg et al., 1988]. In the first of these articles, [Blonder et al., 1982], a method is presented to describe normal-superconducting (NS ) interfaces. The different allowed processes are identified and labelled with their corresponding probabilities as follows: Andreev reflection A(E), normal reflection B(E), and transmission T (E) (cf. Figure 1). These quantities are found by matching the wave functions from either side across the interface and of course A(E) + B(E) + T (E) = 1 must hold, due to the Figure 1. A schematic layout of our Josephson junction. Note that we consider the symmetric case here, hence the reflection and transmission probabilities are the same on both sides. This is not the case in the more general, but also more complicated case of an asymmetric Josephson junction. 124 WDS'08 Proceedings of Contributed Papers, Part III, 124–129, 2008. ISBN 978-80-7378-067-8 © MATFYZPRESS