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arXiv:0707.3897 · Physical Review B

Josephson effect through a quantum dot array

F. S. Bergeret, A. Levy Yeyati, A. Martín‐Rodero

Published Nov 15, 2007• 33 citations
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Abstract

We analyze the ground state properties of an array of quantum dots connected in series between superconducting electrodes. This system is represented by a finite Hubbard chain coupled at both ends to BCS superconductors. The ground state is obtained using the Lanczos algorithm within a low energy theory in which the bulk superconductors are replaced by effective local pairing potentials. We study the conditions for the inversion of the sign of the Josephson coupling ($\ensuremath{\pi}$-junction behavior) as a function of the model parameters. Results are presented in the form of phase diagrams, which provide a direct overall view of the general trends as the size of the system is increased, exhibiting a strong even-odd effect. The analysis of the spin-spin correlation functions and local charges give further insight into the nature of the ground state and how it is transformed by the presence of superconductivity in the leads. Finally, we study the scaling of the Josephson current with the system size and relate these results with previous calculations of Josephson transport through a Luttinger liquid.

DOI
10.1103/physrevb.76.174510

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We present the quantum phase diagram of a superconducting quantum dots array. Superconducting quantum dots are connected through Josephson junctions. We study the effect of finite-range Josephson coupling and also finite-range interactions between the dots. Our field-theoretical study predicts different kinds of phase near the charge degeneracy point. We predict two different kinds of repulsive Luttinger liquid phase and also the possibilities of different density wave instabilities from the inter-play of long-range interaction between the dots and Cooper pair density. We observe different multicritical points with interesting phases.
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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.
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