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Physical Review Letters

Combined effect of Zener and quasiparticle transitions on the dynamics of mesoscopic Josephson junctions

Kieran J. Mullen, Eshel Ben‐Jacob, Z. Schuss

Published Mar 14, 1988• 42 citations
Publisher

Abstract

We study ultrasmall Josephson junctions using the coherent picture to describe the pair dynamics and the semiclassical approach for the quasiparticles. We find that the dynamics depends crucially on the ratio of the superconducting gap energy \ensuremath{\Delta}(T) to the charging energy ${\mathrm{e}}^{2}$/2C. The pair ``Bloch oscillations'' can be practically observed when 2\ensuremath{\Delta}(T)>${\mathrm{e}}^{2}$/2C; for 2\ensuremath{\Delta}(T)${\mathrm{e}}^{2}$/2C the ${\mathrm{I}}_{\mathrm{dc}/\mathrm{e}}$ oscillations due to quasiparticle tunneling can be observed. We discuss under what conditions the I-V characteristic will be S-shaped, exhibiting both oscillations at different values of the current.

DOI
10.1103/physrevlett.60.1097

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Physical review. B, Condensed matter•Dec 1, 1989•5 citations
We study ultrasmall Josephson junctions using the coherent picture to describe the pair dynamics and the semiclassical approach for the quasiparticles. We find that the dynamics depends crucially on the ratio of the superconducting gap energy \ensuremath{\Delta}(T) to the charging energy e2{\mathrm{e}}^{2}e2/2C. The pair ``Bloch oscillations'' can be practically observed when 2\ensuremath{\Delta}(T)>e2{\mathrm{e}}^{2}e2/2C; for 2\ensuremath{\Delta}(T)e2{\mathrm{e}}^{2}e2/2C the Idc/e{\mathrm{I}}_{\mathrm{dc}/\mathrm{e}}Idc/e​ oscillations due to quasiparticle tunneling can be observed. We discuss under what conditions the I-V characteristic will be S-shaped, exhibiting both oscillations at different values of the current.
Within a semiclassical approximation and at the weak-coupling limit, where the charging energy dominates the Josephson coupling energy, the dynamics of small-capacitance superconducting tunnel junctions can be described by a stochastic master equation for the probability distribution of junction charges. We develop a method of solving this equation and obtain an analytic solution when Cooper-pair, quasiparticle, and Zener tunneling are all present under an arbitrary Ohmic conductance at zero temperature. We use this solution to systematically investigate the static and dynamic properties of small Josephson junctions. The charge distributions, current-voltage characteristics, and voltage spectra are analytically calculated and numerically evaluated for wide ranges of parameters. The separate contributions from quasiparticle and Cooper-pair tunneling to voltage spectra are evaluated and compared based on their respective sum rules. The linewidths of the resonance peaks for single-electron tunneling and Bloch oscillations are also discussed.
Two-dimensional arrays of very-small-capacitance Josephson junctions have been studied. At low temperatures the arrays show a transition from superconducting to insulating behavior when the ratio of charging energy to Josephson-coupling energy exceeds the value 1. Insulating behavior coincides withthe occurrence of a charging gap inside the BCS gap, with an S-shaped I-V characteristic. This so far unobserved S shape is predicted to arise from macroscopic quantum coherent effects including Bloch oscillations.
We analyze how the dynamics in a current-biased mesoscopic Josephson junction can be probed by an additional smaller Josephson junction (probe). The probe is connected to a small island in between the Josephson junction and a resistor R >> R Q = h /4e 2 . The current through the resistor results in an effective current bias to the larger JJ. We model the steady state properties of the system under thermal and quantum fluctuations of the feed current. We use a density matrix approach based on the Born-Markov equations treating the current fluctuations perturbatively. The probe is also treated perturbatively assuming the regime of incoherent Cooper-pair tunneling. We find that the I - V characteristics across the probe shows traces of energy bands, Bloch oscillations and Zener tunneling in the current-biased Josephson junction.
We observe that the retrapping current of high-resistance, small-capacitance, superconducting tunnel junctions decreases sharply with decreasing temperature, and then has a puzzling, flat, nonzero minimum at low T, rather than continuing to decrease as quasiparticle damping freezes out. We explain the crossover by considering the combined admittance of the leads and of the junction (including the onset of pair-breaking tunneling at the gap voltage) at both dc and the Josephson frequency. Our analog simulations and analytic approximations both agree well with the data.
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.
Quantum fluctuations of the charge at a current-biased Josephson junction and of the Josephson current couple to the quasiparticle tunneling when the system is at equilibrium in the zero-voltage state. The effect of reducing the capacitance of the junction is studied by means of a model Hamiltonian. When this becomes smaller, charge fluctuations are reduced, but the effective dissipation is also lowered so that the junction displays an underdamped behavior. The Josephson branch is shifted at the voltage e/2C because of charging due to the quasiparticles. A path-integral formulation of the model leads to a high dynamical subgap resistance at I\ensuremath{\rightarrow}0, as recently found experimentally.