Alternating-current properties of short Josephson weak links

Abstract

We calculate the admittance of two types of Josephson weak links-the first is a one-dimensional superconducting wire with a local suppression of the order parameter, and the second is a short S-c-S structure, where S denotes a superconducting reservoir and c is a constriction. The systems of the first type are analyzed on the basis of time-dependent Ginzburg-Landau equations derived by Gor'kov and Eliashberg for gapless superconductors with paramagnetic impurities. It is shown that the impedance Z (Ω) has a maximum as a function of the frequency Ω, and the electric field E Ω is determined by two gauge-invariant quantities. One of them is the condensate momentum Q Ω and another is a potential µ related to charge imbalance. The structures of the second type are studied on the basis of microscopic equations for quasiclassical Green's functions in the Keldysh technique. For short S-c-S contacts (the Thouless energy E Th = D/L 2 ≫ ∆) we present a formula for admittance Y valid frequencies Ω and temperatures T less than the Thouless energy E Th (ħΩ,T ≪ E Th ) but arbitrary with respect to the energy gap ∆. It is shown that, at low temperatures, the absorption is absent [Re(Y ) = 0] if the frequency does not exceed the energy gap in the center of the constriction (Ω < ∆ cos ϕ 0 , where 2ϕ 0 is the phase difference between the S reservoirs). The absorption gradually increases with increasing the difference (Ω − ∆ cos ϕ 0 ) if 2ϕ 0 is less than the phase difference 2ϕ c corresponding to the critical Josephson current. In the interval 2ϕ c < 2ϕ 0 < π, the absorption has a maximum. This interval of the phase difference is achievable in phase-biased Josephson junctions. Close to T c the admittance has a maximum at low Ω which is described by an analytical formula.

Static and dynamic properties of Josephson weak links with singlet and triplet coupling

“…( 31), (32), and (34), look similar to those in our previous publication Ref. 20, they differ essentially because the integrands in corresponding equations are different. 77 In the case of triplet coupling, a little enhancement of Y ′ (Ω) remains only if a characteristic exchange field h is small compared with a subgap ∆ sg .…”

“…Unlike a similar system with perfectly penetrable interfaces considered in Ref. 19,20 , in our case there are barriers at the S/N interfaces so that the interfaces have a finite resistance R if ≡ R . Moreover, we assume that the interface resistance is much larger than the resistance of the N layer, R L = 2L/σ, i. e., the inequality…”

“…We employ the same basic equations for the quasiclassical matrix Green's functions ǧ as in Ref. 20. In the considered onedimensional geometry and in the diffusive limit they obey a generalized Usadel equation of the form 30,[78][79][80][81]…”

“…22 and 23. In all these papers, an anomalous enhancement of the real part of admit-tance Y ′ (Ω) at low frequencies Ω has been obtained. [19][20][21][22][23] The enhancement is caused by quasiparticles with energies in the interval ∆ sg < ǫ < ∆, where the subgap ∆ sg depends on the dc phase difference 2χ 0 .…”