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arXiv:cond-mat/0001375 · Physical review. B, Condensed matter

Andreev scattering and Josephson current in a one-dimensional electron liquid

Ian K. Affleck, Jean-Sébastien Caux, Alexandre M. Zagoskin

Published Jul 1, 2000• 116 citations
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Abstract

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.

DOI
10.1103/physrevb.62.1433

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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≪\ensuremathΔ,(E\ensuremath{\ll}\ensuremath{\Delta},(E\ensuremath≪\ensuremathΔ, 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.
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.
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.
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.
Landau Fermi liquid theory is a fixed-point theory of metals that includes the forward-scattering amplitudes as exact marginal couplings. However, the fixed-point theory that only includes the strict forward scatterings is nonlocal in real space. In this paper, we revisit the Fermi liquid theory for charge-neutral fermions using the field-theoretic functional renormalization group formalism and show how the scale-invariant fixed point emerges as a local theory, which includes not only the forward scatterings but also nonforward scatterings with small but nonzero momentum transfers. In the low-energy limit, the near-forward scattering and pairing interactions take scale-invariant forms if the momentum transfer and the center-of-mass momentum of Cooper pairs, respectively, are comparable to the energy. The coupling functions fully capture the universal low-energy dynamics of the collective modes and instabilities of Fermi liquids. A runaway flow of the coupling function in the particle-hole channel beyond a critical interaction suggests an instability toward an ordered phase with a wave vector that depends on the interaction strength. At the critical interaction, the instability corresponds to the uniform Pomeranchuk or Stoner instability, but the momentum of the leading instability becomes nonzero for stronger attractive interaction. In the particle-particle channel, the coupling function reveals the dynamics of the unstable mode associated with the BCS instability. When an unstable normal metal evolves into the superconducting state, there exists a period in which a superconducting state with spatially nonuniform phase appears due to the presence of unstable Cooperon modes with nonzero momenta.
We study junctions of single-channel spinless Luttinger liquids using bosonization. We generalize earlier studies by allowing the junction to be superconducting and find new charge nonconserving low-energy fixed points. We establish the existence of $g\ensuremath{\leftrightarrow}1/g$ duality (where ggg is the Luttinger liquid parameter) between the charge-conserving (normal) junction and the charge nonconserving (superconducting) junction by evaluating and comparing the scaling dimensions of various operators around the fixed points in both the normal and superconducting sectors of the theory. For the most general two-wire junction, we show that there are two conformally invariant one-parameter families of fixed points which are also connected by a duality transformation. We also show that the stable fixed point for the two-wire superconducting junction corresponds to the situation where the crossed Andreev reflection (an incoming electron is transmitted as an outgoing hole) is perfect between the wires. For the three-wire junction, we study, in particular, the superconducting analogs of the chiral ${\mathbsf{D}}_{\mathbsf{P}}$ and the disconnected fixed points obtained earlier in the literature in the context of charge-conserving three-wire junctions. We show that these fixed points can be stabilized for g<1g<1g<1 (repulsive electrons) within the superconducting sector of the theory which makes them experimentally relevant.