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Journal of Applied Physics

Influence of the current-phase relation on the critical-current-applied-magnetic-flux dependence in parallel-connected Josephson junctions

Won−Tien Tsang, T. Van Duzer

Published Jun 1, 1976• 10 citations
Publisher

Abstract

The form of the current-phase relations for the Josephson junctions is shown to have a significant influence on the relation Ic(ϑa) between critical current and applied flux for two junctions connected in parallel in a superconducting circuit. The observed one-flux-quantum periodicity and inversion symmetry of the Ic(ϑa) relation are shown to result from the fact that the current-phase, i-φ, relations of the junctions satisfy i (φ+2mπ) =i (φ) and i (−φ) =−i (φ), respectively. It is also shown that if the current-phase relations for the two junctions are different, an asymmetry appears in the Ic(ϑa).

DOI
10.1063/1.322986

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The form of the current-phase relations for the Josephson junctions is shown to have a significant influence on the relation Ic(ϑa) between critical current and applied flux for two junctions connected in parallel in a superconducting circuit. The observed one-flux-quantum periodicity and inversion symmetry of the Ic(ϑa) relation are shown to result from the fact that the current-phase, i-φ, relations of the junctions satisfy i (φ+2mπ) =i (φ) and i (−φ) =−i (φ), respectively. It is also shown that if the current-phase relations for the two junctions are different, an asymmetry appears in the Ic(ϑa).
We study a 0-$\ensuremath{\pi}$ dc superconducting quantum interference device (SQUID) with asymmetric inductances and critical currents of the two Josephson junctions (JJs). By considering such a dc SQUID as a black box with two terminals, we calculate its effective current-phase relation ${I}_{s}(\ensuremath{\psi})$ and the Josephson energy $U(\ensuremath{\psi})$, where $\ensuremath{\psi}$ is the Josephson phase across the terminals. We show that there is a domain of parameters where the black box has the properties of a $\ensuremath{\varphi}$ JJ with degenerate ground state phases $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$. The $\ensuremath{\varphi}$ domain is rather large, so one can easily construct a $\ensuremath{\varphi}$ JJ experimentally. We derive the current phase relation and show that it can be tuned in situ by applying an external magnetic flux resulting in a continuous transition between the systems with static solutions $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi},\phantom{\rule{0.16em}{0ex}}\ensuremath{\psi}={\ensuremath{\varphi}}_{0} ({\ensuremath{\varphi}}_{0}\ensuremath{\ne}0,\ensuremath{\pi})$ and even $\ensuremath{\psi}={\ensuremath{\varphi}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$. The dependence of ${\ensuremath{\varphi}}_{0}$ on applied magnetic flux is not $2\ensuremath{\pi}$ (one flux quantum) periodic.
Abstract In this study, we investigate the quantum phase transition in a Josephson junction consisting of parallel-connected quantum dots (QDs) with a magnetic flux φ B threading the central region. Employing the surrogate model approach, we discretize the superconducting (SC) electrode into three discrete energy levels, which enables direct diagonalization of the Hamiltonian to determine the system’s ground state. Our calculations demonstrate that when Coulomb interaction is present in only one QD, the system exhibits a phase transition between singlet and doublet states. The magnetic flux exerts a minor influence on the singlet state while substantially affecting the doublet state. When both QDs possess Coulomb interactions, the system undergoes two successive phase transitions as the SC phase difference increases: the ground state evolves from a doublet to a singlet and subsequently to a triplet state at ϕ = π . Enhancement of the magnetic flux suppresses the doublet and triplet phases, ultimately stabilizing the singlet state. Within this regime, increasing the Coulomb interaction strength U does not trigger a singlet-triplet transition; rather, it induces a transition between upper and lower singlet states, resulting in a pronounced peak in the critical current as U increases. Finally, we explore the scenario where the tunneling self-energy Γ between the QDs and SCs exceeds the SC pairing potential Δ. In this case, doublet states prevail, and the system manifests a phase transition exclusively between doublet and triplet states when ϕ B = 0 . Upon application of a magnetic flux, all three states converge, giving rise to a triple point in the ( φ , φ B ) parameter space.
A detailed review of theories describing the current-phase relations in Josephson junctions based on heterostructures with ferromagnetic layers is presented. The particular attention has been focused on the possibilities of making the so-called ϕ-junctions in which the ground state in the absence of the current is realized when the Josephson phase ϕ is nonzero. The recently popular speculations concerning the possibility of applying the Josephson π- and ϕ-junctions, which are made on the basis of heterostructures with ferromagnetic layers, to the formation of quantum bits (qubits) have been illustrated. An attempt to formulate the requirements for the characteristics of the Josephson heterostructures based on the first principles of the quantum-mechanical description of superconducting interferometers with a low inductance has been made.
We have calculated the total critical current as a function of applied magnetic flux for a superconducting interferometer consisting of many Josephson junctions in parallel. An enhancement and narrowing of the periodic principal maxima in the critical current versus flux characteristic is predicted as the number of junctions in parallel increases, even when the effects of finite self- and mutual inductances and nonuniformity of the junction critical currents are included in the calculations. The possible application of the superconducting quantum interference grating, or SQUIG, for the detection of magnetic flux is discussed.
This review provides a theoretical basis for understanding the current-phase relation (C\ensuremath{\Phi}R) for the stationary (dc) Josephson effect in various types of superconducting junctions. The authors summarize recent theoretical developments with an emphasis on the fundamental physical mechanisms of the deviations of the C\ensuremath{\Phi}R from the standard sinusoidal form. A new experimental tool for measuring the C\ensuremath{\Phi}R is described and its practical applications are discussed. The method allows one to measure the electrical currents in Josephson junctions with a small coupling energy as compared to the thermal energy. A number of examples illustrate the importance of the C\ensuremath{\Phi}R measurements for both fundamental physics and applications.
Conventional superconductor/insulator/superconductor (SIS) Josephson junctions, devices where two superconductors are separated by a tunnel barrier are technologically important as elements in quantum circuits, particularly with their key role in superconducting qubits. An important characteristic of Josephson junctions is the relation between the supercurrent Is and the phase difference ϕϕϕ between them. For SIS junctions, the current-phase relation is sinusioidal and 2πππ periodic. Other types of Josephson junctions, where the material between the superconductors is a weak link or a normal metal (N) may have non-sinusoidal current-phase relations that are still 2πππ periodic. We show here that a multi-terminal diffusive SNS Josephson junction with 4 superconducting contacts can show a current phase relation between two of the contacts that is a superposition of 2πππ and 4πππ periodic components whose relative strength is controlled by the phase difference between the other two contacts, becoming 2πππ or 4πππ periodic for certain values of this phase difference. This tunability might have applications in tailoring the Hamiltonians of superconducting quantum circuits.