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arXiv:cond-mat/0109103 · Physical Review Letters

Josephson Current in Luttinger Liquid-Superconductor Junctions

Jean-Sébastien Caux, Hubert Saleur, F. Siano

Published Feb 21, 2002• 53 citations
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Abstract

We study the Josephson current through a Luttinger liquid in contact with two superconductors. We show that it can be deduced from the Casimir energy in a two-boundary version of the sine-Gordon model. We develop a new thermodynamic Bethe ansatz, which, combined with a subtle analytic continuation procedure, allows us to calculate this energy in closed form, and obtain the complete current-crossover function from the case of complete normal to complete Andreev reflection.

DOI
10.1103/physrevlett.88.106402

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•Aug 23, 2021•0 citations
We study the Josephson current through a Luttinger liquid in contact with two superconductors. We show that it can be deduced from the Casimir energy in a two-boundary version of the sine-Gordon model. We develop a new thermodynamic Bethe ansatz, which, combined with a subtle analytic continuation procedure, allows us to calculate this energy in closed form, and obtain the complete current-crossover function from the case of complete normal to complete Andreev reflection.
A theory describing a one-dimensional Luttinger liquid in contact with a superconductor is developed. Boundary conditions for the fermion fields describing Andreev reflection at the contacts are derived and used to construct a bosonic representation of the fermions. The Josephson current through a superconductor/Luttinger liquid/superconductor junction is considered for both perfectly and poorly transmitting interfaces. In the former case, the Josephson current at low temperatures is found to be essentially unaffected by electron-electron interactions. In the latter case, significant renormalization of the Josephson current occurs. The profile of the (induced) condensate wave function in a semi-infinite Luttinger liquid in contact with a superconductor is shown to decay as a power law, the exponent depending on the sign and strength of the interactions. In the case of repulsive (attractive) interactions the decay is faster (slower) than in their absence. An equivalent method of calculating the Josephson current through a Luttinger liquid, which employs the bosonization of the system as a whole (i.e., superconductor, as well as Luttinger liquid) is developed and shown to give the results equivalent to those obtained via boundary conditions describing Andreev reflection. \textcopyright{} 1996 The American Physical Society.
Andreev scattering and the Josephson current through a one-dimensional interacting electron liquid sandwiched between two superconductors are reexamined. We first present some apparently new results on the noninteracting case by studying an exactly solvable tight-binding model rather than the usual continuum model. We show that perfect Andreev scattering (i.e., zero normal scattering) at the Fermi energy can only be achieved by fine-tuning junction parameters, a fine-tuning which is possible even with bandwidth mismatch between superconductor and normal metal. We also obtain exact results for the Josephson current, which is generally a smooth function of the superconducting phase difference except when the junction parameters are adjusted to give perfect Andreev scattering, in which case it becomes a sawtooth function. We then observe that, even when interactions are included, all low-energy properties of a junction $(E\ensuremath{\ll}\ensuremath{\Delta},$ the superconducting gap) can be obtained by ``integrating out'' the superconducting electrons to obtain an effective Hamiltonian describing the metallic electrons only with a boundary pairing interaction. This boundary model provides a suitable starting point for bosonization-renormalization group-boundary conformal field theory analysis. We argue that total normal reflection and total Andreev reflection correspond to two fixed points of the boundary renormalization group. For repulsive bulk interactions the Andreev fixed point is unstable and the normal one stable. However, the reverse is true for attractive interactions. This implies that a generic junction Hamiltonian (without fine-tuned junction parameters) will renormalize to the normal fixed point for repulsive interations but to the Andreev one for attractive interations. An exact mapping of our tight-binding model to the Hubbard model with a transverse magnetic field is used to help understand this behavior. We calculate the critical exponents, which are different at these two different fixed points.
The Josephson current through a one-dimensional (1D) interacting electron system of length L coupled with two superconductors is studied in the presence of a barrier potential at the junction interfaces. The 1D system is treated within the framework of the Tomonaga-Luttinger liquid, in which the interaction is characterized by correlation exponents K ρ and K σ . We pay attention to the influence of Andreev reflection, which modifies the low-energy property of the 1D electron system near the junction interfaces. It is found that the critical current at low temperatures decays with increasing L as j c ∝(1/ L ) 2/ K ρ -1 , which is in striking contrast to the previous result j c ∝(1/ L ) 1/ K ρ + K σ -1 , obtained by Fazio et al. taking no account of the influence of Andreev reflection.
We study the Josephson effect through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. The interactions are treated in the framework of the Luttinger model. At zero temperature, the Josephson critical current decays algebraically with increasing distance between the junctions. In a Luttinger liquid of finite size, the Josephson current depends on the total number of electrons modulo 4. These parity effects are studied for a ring, coupled capacitively to a gate voltage and threaded by a magnetic flux. Interactions introduce qualitatively new features compared to the noninteracting case.
We calculate both the dc and the ac Josephson current through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. We treat the (repulsive) Coulomb interaction in the framework of the one-channel, spin-1/2 Luttinger model. The Josephson current is obtained for two geometries of experimental relevance: a quantum wire and a ring. At zero temperature, the critical current is found to decay algebraically with increasing distance d between the junctions. The decay is characterized by an exponent which depends on the strength of the interaction. At finite temperatures T, lower than the superconducting transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​, there is a crossover from algebraic to exponential decay of the critical current as a function of d, at a distance of the order of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/kBT{\mathit{k}}_{\mathit{BT}}kBT​. Moreover, the dependence of critical current on temperature shows nonmonotonic behavior. If the Luttinger liquid is confined to a ring of circumference L, coupled capacitively to a gate voltage and threaded by a magnetic flux, the Josephson current shows remarkable parity effects under the variation of these parameters. For some values of the gate voltage and applied flux, the ring acts as a \ensuremath{\pi} junction. These features are robust against thermal fluctuations up to temperatures on the order of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/kBL{\mathit{k}}_{\mathit{BL}}kBL​. For the wire geometry, we have also studied the ac-Josephson effect. The amplitude and the phase of the time-dependent Josephson current are affected by electron-electron interactions. Specifically, the amplitude shows pronounced oscillations as a function of the bias voltage due to the difference between the velocities of spin and charge excitations in the Luttinger liquid. Therefore, the ac-Josephson effect can be used as a tool for the observation of spin-charge separation. \textcopyright{} 1996 The American Physical Society.
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.