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arXiv:cond-mat/9510059 · Physical review. B, Condensed matter

One-dimensional Josephson arrays as superlattices for single Cooper pairs

A. A. Odintsov

Published Jul 1, 1996• 18 citations
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Abstract

We investigate uniform one-dimensional arrays of small Josephson junctions [${\mathit{E}}_{\mathit{J}}$\ensuremath{\ll}${\mathit{E}}_{\mathit{C}}$, ${\mathit{E}}_{\mathit{C}}$=(2e${)}^{2}$/2C] with a realistic Coulomb interaction U(x)=${\mathit{E}}_{\mathit{C}}$\ensuremath{\lambda}exp(-|x|/\ensuremath{\lambda}) (here \ensuremath{\lambda}\ensuremath{\gg}1 is the screening length in units of the lattice constant of the array). At low energies this system can be described in terms of interacting Bose particles (extra single Cooper pairs) on the lattice. With increasing concentration \ensuremath{\nu} of extra Cooper pairs, a crossover from the Bose gas phase to the Wigner crystal phase and then to the superlattice regime occurs. The phase diagram in the superlattice regime consists of commensurable insulating phases with \ensuremath{\nu}=1/l (l is integer) separated by superconducting regions where the current is carried by excitations with fractional electric charge q=\ifmmode\pm\else\textpm\fi{}2e/l. The Josephson current through a ring-shaped array pierced by magnetic flux is calculated for all of the phases. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.1228

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We investigate uniform one-dimensional arrays of small Josephson junctions [EJ{\mathit{E}}_{\mathit{J}}EJ​\ensuremath{\ll}EC{\mathit{E}}_{\mathit{C}}EC​, EC{\mathit{E}}_{\mathit{C}}EC​=(2e)2{)}^{2})2/2C] with a realistic Coulomb interaction U(x)=EC{\mathit{E}}_{\mathit{C}}EC​\ensuremath{\lambda}exp(-|x|/\ensuremath{\lambda}) (here \ensuremath{\lambda}\ensuremath{\gg}1 is the screening length in units of the lattice constant of the array). At low energies this system can be described in terms of interacting Bose particles (extra single Cooper pairs) on the lattice. With increasing concentration \ensuremath{\nu} of extra Cooper pairs, a crossover from the Bose gas phase to the Wigner crystal phase and then to the superlattice regime occurs. The phase diagram in the superlattice regime consists of commensurable insulating phases with \ensuremath{\nu}=1/l (l is integer) separated by superconducting regions where the current is carried by excitations with fractional electric charge q=\ifmmode\pm\else\textpm\fi{}2e/l. The Josephson current through a ring-shaped array pierced by magnetic flux is calculated for all of the phases. \textcopyright{} 1996 The American Physical Society.
We have measured the current-voltage $(I\ensuremath{-}V)$ characteristics of small-capacitance single Josephson junctions at low temperatures (T=0.02−−0.6K),(T=0.02--0.6\mathrm{K}),(T=0.02−−0.6K), where the strength of the coupling between the single junction and the electromagnetic environment was controlled with one-dimensional arrays of dc superconducting quantum interence devices (SQUIDs). The single-junction $I\ensuremath{-}V$ curve is sensitive to the impedance of the environment, which can be tuned in situ. We have observed Coulomb blockade of Cooper-pair tunneling and even a region of negative differential resistance, where the zero-bias resistance ${R}_{0}^{\ensuremath{'}}$ of the SQUID arrays is much higher than the quantum resistance ${R}_{K}\ensuremath{\equiv}{h/e}^{2}\ensuremath{\approx}26\mathrm{k}\ensuremath{\Omega}.$ The negative differential resistance is evidence of the coherent single-Cooper-pair tunneling within the theory of current-biased single Josephson junctions. Based on this theory, we have calculated the $I\ensuremath{-}V$ curves numerically in order to compare then with the experimental ones at ${R}_{0}^{\ensuremath{'}}\ensuremath{\gg}{R}_{K}.$ The numerical calculation agrees with the experiments qualitatively. We also discuss the ${R}_{0}^{\ensuremath{'}}$ dependence of the single-Josephson-junction $I\ensuremath{-}V$ curve in terms of the superconductor-insulator transition driven by changing the coupling to the environment.
We review experiments on small-capacitance Josephson junctions. When the Josephson junction is fabricated in the configuration of dc superconducting quantum interference device (SQUID), the Josephson coupling can be tuned IN SITU with an external magnetic field. The electrical transport properties of one-dimensional arrays of small-capacitance dc SQUIDs have been investigated in order to study the superconductor-insulator transition. The arrays have also been used to bias a single Josephson junction, and a clear Coulomb blockade of Cooper-pair tunneling has been observed in the single junction.
We have measured the current-voltage characteristics of small-capacitance single Josephson junctions at low temperatures ( T< or =0.04 K), where the strength of the coupling between the single junction and the electromagnetic environment was controlled with one-dimensional arrays of dc SQUIDs. We have clearly observed Coulomb blockade of Cooper-pair tunneling and even a region of negative differential resistance, when the zero-bias resistance of the SQUID arrays is much higher than the quantum resistance h/e(2) approximately 26 kOmega. The negative differential resistance is evidence of coherent single-Cooper-pair tunneling in the single Josephson junction.
The phase diagram of a two-dimensional Josephson array of mesoscopic objects (superconducting granules, superfluid helium in a porous medium, traps with Bose-condensed atoms, etc.) is examined. Quantum fluctuations in both the modulus and phase of the superconducting order parameter are taken into account within a lattice boson Hubbard model. Modulating the average occupation number n 0 of the sites in the system (the “number of Cooper pairs” per granule, the number of atoms in a trap, etc.) leads to changes in the state of the array, and the character of these changes depends significantly on the region of the phase diagram being examined. In the region where there are large quantum fluctuations in the phase of the superconducting order parameter, variation of the chemical potential causes oscillations with alternating superconducting (superfluid) and normal states of the array. On the other hand, in the region where the bosons interact weakly, the properties of the system depend monotonically on n 0 . Lowering the temperature and increasing the particle interaction force lead to a reduction in the width of the region of variation in n 0 within which the system properties depend weakly on the average occupation number. The phase diagram of the array is obtained by mapping this quantum system onto a classical two-dimensional XY model with a renormalized Josephson coupling constant and is consistent with our quantum path-integral Monte Carlo calculations.
Superconductivity mediated by Cooper quartets of charge 4e is a phenomenon of key importance for both the understanding of certain exotic superconductors and the engineering of quantum memories protected at the hardware level by topological order. Josephson junction arrays in the shape of the dice lattice constitute one of the main candidates for the realization of this phase of matter. Here, we analyze numerical signatures of the emergence of this exotic phase when superconducting dice arrays are frustrated by inserting one third of a flux quantum per rhombic plaquette. We adopt simulations of relaxation dynamics to study the critical current of such devices and analyse the Fourier decomposition of their bulk supercurrents. A further characterization of these systems at finite temperature is obtained through Monte Carlo techniques and two-dimensional infinite tensor networks. Our results indicate that the observed peaks of the critical current and temperature at frustration 1/3 correspond to a superconductor-insulator phase transition compatible with the deconfinement of half-vortices, while the low-temperature correlation functions of the model confirm the onset of a 4e superconducting phase mediated by Cooper quartets. We finally address the effects of Josephson energy and flux disorder typical of experimental arrays, and comment on the role of charging energies in the corresponding two-dimensional quantum model.
We have investigated how the transport properties of small-Josephson-junction arrays vary depending on a quantum control parameter (the ratio of the Josephson coupling energy E J to the charging energy E C ), the magnitude of dissipation, and the dimensionality d of the system. We found that the arrays show Coulomb blockade behavior when R S is larger than about d R Q ( R Q ≡ h /4 e 2 =6.45 kΩ) and E J / E C is smaller than a threshold depending on the value of R S and the dimensionality. The result is discussed in terms of quantum phase transition induced by the charging effect and the dissipation.