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Reviews of Modern Physics

The current-phase relation in Josephson junctions

A. A. Golubov, M. Yu. Kupriyanov, Evgeni V. Il’ichev

Published Apr 26, 2004• 1,427 citations
Publisher

Abstract

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.

DOI
10.1103/revmodphys.76.411

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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.
Purpose Josephson junctions act as active elements in superconducting electronics. The behavior of this nonlinear element is characterized by the relation between current and the quantum mechanical phase‐difference. For an accurate device modeling, detailed knowledge about this relation is necessary. This paper aims to discuss these issues. Design/methodology/approach To obtain detailed information, a method for DC measurement of the current‐phase relation suitable for all kinds of superconducting circuit elements was accomplished. Findings The authors developed a linear transformation algorithm to calculate the current‐phase relation from the measured data. Research limitations/implications It turns out that in future designs additional connections and special test structures are required to gain more knowledge about inductance values required for the algorithm. Originality/value Based on the inverse calculation of that algorithm, the authors found a 7 percent deviation of the current‐phase relation of a standard superconductor/insulator/superconductor Josephson junction from the predicted sine‐wave behavior. Furthermore, the paper suggests to use this method to evaluate the current‐phase relation of new Josephson elements such as a superconductor/ferromagnet/superconductor junction. Therefore, the authors will deposit the new element directly on the chip with the test setup fabricated with standard Nb‐technology.
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.
The modern physics of superconductivity can be called the physics of unconventional superconductivity. The discovery of the d-wave symmetry of the order parameter in high-temperature superconductors and the triplet superconductivity in compound Sr2RuO4 has caused a huge stream of theoretical and experimental investigations of unconventional superconductors. In this review we discuss some novel aspects of the Josephson effect which are related to the symmetry of the order parameter. The most intriguing of them is spontaneous current generation in an unconventional weak link. The example of a Josephson junction in the form of a grain boundary between two disorientated d-wave or f-wave superconductors is considered in detail. Josephson current–phase relations and the phase dependences of the spontaneous current that flows along the interface are analyzed. The spontaneous current and spontaneous phase difference are manifestations of the time-reversal symmetry (𝒯 ) breaking states in the system. We analyzed the region of appearance of 𝒯-breaking states as function of temperature and mismatch angle. A review of the basics of superconducting qubits with emphasis on specific properties of d-wave qubits is given. Recent results in the problem of decoherence in d-wave qubits, which is the major concern for any qubit realization, are presented.
Properties of weak links between two superconductors, or Josephson junctions, make them interesting for fundamental physics research. Since their discovery over four decades ago, they have provided a unique way to study the behavior of the superconducting quantum phase. More recently, ultra small, or mesoscopic, Josephson junctions with substantial single Cooper pair charging energy have gained interest due to their behavior as macroscopic quantum objects. In addition to the theoretical interest, Josephson junctions can be used as active elements in circuit applications. Particularly, in this Thesis we study two different devices. We develop the required theoretical treatments, derive device properties, and compare the results with experimental data. The first application is a Josephson voltage standard based on externally damped Superconductor - Insulator - Superconductor junctions. It consists of an array of large Josephson junctions connected in series and irradiated with a 70 GHz microwave signal. Phase locking the Josephson dynamics into the signal leads to the quantization of the voltage. This is utilized in metrology. We introduce a new circuit solution based on frequency dependent damping of the junctions. Optimization and some designs for practical arrays are presented. The purpose is to find such a design that the array is fast, has low power consumption and is as stable as possible. Arrays able to generate DC voltages of order 1 volt with metrological accuracy are demonstrated experimentally and their applicability in AC voltage calibrations is analyzed. The second application is the Bloch Oscillating Transistor (BOT). The BOT is based on controlling the Cooper pair current in an ultra small Josephson junction by means of quasiparticles tunneling through a normal junction. As part of the thesis work, the principle of operation is first demonstrated computationally. The model is then refined to yield quantitative predictions of the characteristics. Finally, an analytic theory for the device is developed and the properties as an amplifier are derived.
Deep theoretical understanding of the electrical response of Josephson junctions is indispensable regarding both recent discoveries of new kinds of superconductivity and technological advances such as superconducting quantum computers. Here, we study the microscopic theory of the dc current-biased I-V characteristics of Josephson tunnel junctions. We derive an analytical formula of the I-V characteristics of generic junctions. We identify subharmonics of the I-V characteristics and their underlying mechanism as the feedback effect of intrinsic ac currents generated by voltage pulses in the past. We apply our theory to analytically solve the Werthamer equation and describe various dc current-biased I-V characteristics as a function of softening of the superconducting gap. Strikingly, we identify voltage staircases of the I-V characteristics in a genuine Josephson junction without ac current bias or qubit dynamics. Our general analytical formalism opens new avenues for a microscopic understanding of I-V characteristics of Josephson junctions that have been limited to phenomenological models so far.
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).