Abstract
We provide a new analytical tool to calculate the energies of Andreev bound states (ABS) in long imperfect SNS junctions, at present these can only be described by numerical tools. We model an NS junction as a delta-function "Andreev" impurity, i.e., a localized potential which scatters an electron into a hole with opposite spin. We show using the scattering matrix formalism that, quite surprisingly, an "Andreev" impurity is equivalent to an NS junction characterized by both Andreev reflection and a finite amount of normal scattering. The ABS energies are then calculated using the T-matrix formalism applied to a system with two Andreev impurities. Our results lie between those for a perfect long SNS junction limit described by the Andreev approximation (ABS energies depend linearly on the phase and are independent of the chemical potential) and the particle-in-the-box limit (bound state energies are independent of the phase and have a linear dependence on the chemical potential). Moreover, we recover a closed-form expression for the ABS energies by expanding around the particle-in-the-box limit.
Highlights
Our results lie between those for a perfect long SNS junction limit described by the Andreev approximation (ABS energies depend linearly on the phase and are independent of the chemical potential) and the particle-in-the-box limit
When there is no normal scattering at the leads, the physics of the Andreev bound states (ABSs) in this limit is described by the Andreev approximation [13,14,15], yielding a linear dependence of the ABS energies on the phase; in this limit, their energies are independent of the chemical potential
We have shown that by introducing two “Andreev” impurities into a normal metal and by employing the T -matrix formalism, we can model a long imperfect SNS junction and the formation of Andreev bound states
Summary
The formation of Andreev bound states (ABSs) [1,2,3,4,5] in long SNS junctions [6,7,8,9] has been approached by analytical tools such as the Bogoliubov–de Gennes equations [10,11,12], the Andreev approximation [13,14,15], as well as various other approaches [16,17,18,19,20,21,22,23]. We will first show that we can model an imperfect NS junction by considering an “Andreev”-type impurity: a δfunction localized potential that scatters an electron into a hole with opposite spin This equivalence is demonstrated using the scattering formalism: we find that there are regimes of parameters in which the values of the reflection and Andreev reflection coefficients in the NS junction can be the same as those generated by an “Andreev” impurity. The resulting Green’s function in the region between the two impurities corresponds to that of a normal region in an SNS junction, and it allows us to describe the formation of ABSs. We calculate the resulting dependence of the ABS energies on various parameters such as the gate voltage, the phase difference between the two SCs, and the normal scattering at the leads, and we show that it is consistent with previous findings This situation describes probably quite accurately the realistic parameters for many experimental NS interfaces
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
