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arXiv:2108.10333 · Physical Review Research

Analytic approach to transport in superconducting junctions with arbitrary carrier density

F. Setiawan, Johannes Hofmann

Published Nov 8, 2022• 14 citations
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Abstract

By developing a general framework for transport across superconducting junctions that goes beyond the standard Andreev approximation, two crucial transport signatures are proposed for $s$-wave Josephson junctions along the BCS-BEC crossover: negative differential conductance as a signature of unitarity and a complete suppression of multiple-Andreev-reflection-induced current as a hallmark of the splitting point.

DOI
10.1103/physrevresearch.4.043087

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By developing a general framework for transport across superconducting junctions that goes beyond the standard Andreev approximation, two crucial transport signatures are proposed for sss-wave Josephson junctions along the BCS-BEC crossover: negative differential conductance as a signature of unitarity and a complete suppression of multiple-Andreev-reflection-induced current as a hallmark of the splitting point.
Transport in Josephson junctions is commonly described using a simplifying assumption called the Andreev approximation, which assumes that excitations are fixed at the Fermi momentum and only Andreev reflections occur at interfaces (with no normal reflections). This approximation is appropriate for BCS-type superconductors, where the chemical potential vastly exceeds the pairing gap, but it breaks down for superconductors with low carrier density, such as topological superconductors, doped semiconductors, or superfluid quantum gases. Here, we present a generic analyticalanalyticalanalytical framework for calculating transport in Josephson junctions that lifts up the requirement of the Andreev approximation. Using this general framework, we study in detail transport in Josephson junctions across the BCS-BEC crossover, which describes the evolution from a BCS-type superconductor with loosely-paired Cooper pairs to a BEC of tighly-paired dimers. As the interaction is tuned from the BCS to the BEC regime, we find that the overall subgap current caused by multiple Andreev reflections decreases, but nonlinearities in the current-voltage characteristic called the subharmonic gap structure become more pronounced near the intermediate unitary limit, giving rise to sharp peaks and dips in the differential conductance with even negativenegativenegative conductance at specific voltages.
We present a theory of SNS junctions, where a normal metal is in contact with two superconductors, along the crossover from the BCS to the BEC regime. Such systems can be realized experimentally by gate-controlled doping of thin film superconductors. We show that Andreev levels exist even in the BEC regime, as long as there is sufficient density of states in the normal metal. We derive the Josephson current through one-dimensional Sharvin contacts. We find that the Josephson current at zero temperature has no peak along the BCS-BEC crossover at T=0KT=0\phantom{\rule{4.pt}{0ex}}\text{K}T=0K unless the order paramter $\mathrm{\ensuremath{\Delta}}$ is peaked, as it is the case for superconductors with one-dimensional density of states. At finite temperature, however, we find that the Josephson current is maximal at the BCS-BEC crossover for any dimension. We identify the mechanism for this phenomenon to be a decrease in the number of Andreev levels, accompanied by an increase in excitation energy to the unoccupied levels, along the BCS-BEC crossover. This makes it less likely that these states are thermally occupied. As a consequence, the Josephson current is more strongly reduced on the BCS side of the crossover, resulting in a maximal Josephson current at the BCS-BEC crossover. Furthermore, we find that the current phase relation changes along the BCE-BEC crossover with a slope which has a pronounced maximum on the BEC-side of the BCS-BEC crossover. This maximum becomes larger and sharper as the temperature is lowered.
One of the challenges in the pursuit of solid state BEC superconductors is the need for unambiguous experimental signatures. Here, the authors propose a new signature, relying on Andreev reflection, which distinguishes BEC from BCS superconductors. They demonstrate that, in a standard tunneling spectroscopy setup, a BEC superconductor exhibits suppressed Andreev reflection; however, if the tunneling barrier becomes a potential well, then appreciable Andreev reflection can be revived. Their predictions can be probed in various architectures that are also discussed.
We develop a microscopic model to describe the Josephson tunneling between two superfluid reservoirs of ultracold fermionic atoms which accounts for the dependence of the critical current on both the barrier height and the interaction strength along the crossover from the BCS to the BEC regime. Building on a previous study [F. Meier and W. Zwerger, Phys. Rev. A 64, 033610 (2001)] of weakly interacting bosons, we derive analytic results for the Josephson critical current at zero temperature for homogeneous and trapped systems at arbitrary coupling. The critical current exhibits a maximum near the unitarity limit which arises from the competition between the increasing condensate fraction and a decrease in the chemical potential along the evolution from the BCS to the BEC limit. Our results agree quantitatively with numerical simulations and recent experimental data.
Although Josephson junctions can be viewed as highly nonlinear impedances for superconducting quantum technologies, they also possess internal dynamics that may strongly affect their behavior. Here, we construct a computational framework that includes a microscopic description of the junction (full fledged treatment of both the superconducting condensate and the quasiparticles) in the presence of a surrounding electrical circuit. Our approach generalizes the standard resistor capacitor Josephson model to arbitrary junctions (including, e.g., multiterminal geometries and/or junctions that embed topological or magnetic elements) and arbitrary electric circuits treated at the classical level. By treating the superconducting condensate and quasiparticles on equal footings, we capture nonequilibrium phenomena such as multiple Andreev reflection. We show that the interplay between the quasiparticle dynamics and the electrical environment leads to the emergence of new phenomena. In a RC circuit connected to single channel Josephson junction, we find out-of-equilibrium current-phase relations that are strongly distorted with respect to the (almost sinusoidal) equilibrium one, revealing the presence of the high harmonic ac Josephson effect. In an RLC circuit connected to a junction, we find that the shape of the resonance is strongly modified by the quasiparticle dynamics: close to resonance, the current can be smaller than without the resonator. Our approach provides a route for the quantitative modeling of superconducting-based circuits.
We have calculated all the components of the current in a short one-dimensional channel between two superconductors for arbitrary voltages and transparencies DDD of the channel. We demonstrate that in the ballistic limit ( $D\ensuremath{\simeq}1$) the crossover between the quasistationary evolution of the Josephson phase difference $\ensuremath{\phi}$ at small voltages and transport by multiple Andreev reflections at larger voltages can be described as the Landau-Zener transition induced by finite reflection in the channel. For perfect transmission and vanishing energy relaxation rates the stationary current-phase relation is never recovered, and $I(\ensuremath{\phi}){\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}I}_{c}\ensuremath{\mid}\mathrm{sin}\ensuremath{\phi}/2\ensuremath{\mid}\mathrm{sgn}V$ for arbitrary small voltages.