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arXiv:cond-mat/9803066 · Physical Review Letters

Coulomb Blockade in Superconducting Quantum Point Contacts

D. V. Averin

Published May 3, 1999• 35 citations
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Abstract

The amplitude of the Coulomb blockade oscillations is calculated for a single-mode Josephson junction with arbitrary electron transparency $D$. It is shown that a mechanism related to chiral anomaly completely suppresses the Coulomb blockade in ballistic junctions with $D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$. At finite reflection probability, the suppression process is described quantitatively in terms of the Landau-Zener transition.

DOI
10.1103/physrevlett.82.3685

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The amplitude of the Coulomb blockade oscillations is calculated for a single-mode Josephson junction with arbitrary electron transparency DDD. It is shown that a mechanism related to chiral anomaly completely suppresses the Coulomb blockade in ballistic junctions with D=1D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1D=1. At finite reflection probability, the suppression process is described quantitatively in terms of the Landau-Zener transition.
We develop a theory of Coulomb oscillations in superconducting devices in the limit of small charging energy EC≪ΔE_C \ll ΔEC​≪Δ. We consider a small superconducting grain of finite capacity connected to two superconducting leads by nearly ballistic single-channel quantum point contacts. The temperature is supposed to be very low, so there are no single-particle excitations on the grain. Then the behavior of the system may be described as quantum mechanics of the superconducting phase on the island. The Josephson energy as a function of this phase has two minima which become degenerate at the phase difference on the leads equal to πππ, the tunneling amplitude between them being controlled by the gate voltage at the grain. We find the Josephson current and its low-frequency fluctuations and predict their periodic dependence on the induced charge Qx=CVgQ_x=C V_gQx​=CVg​ with period 2e2e2e.
We develop a theory of Coulomb oscillations in superconducting devices in the limit of small charging energy E C ≪Δ. We consider a small superconducting grain with finite capacitance connected to two superconducting leads by nearly ballistic single-channel quantum point contacts. The temperature is assumed to be very low, so there are no single-particle excitations on the grain. Then the behavior of the system can be described in terms of the quantum mechanics of the superconducting phase on the island. The Josephson energy as a function of this phase has two minima that become degenerate when the phase difference on the leads equals to π, the tunneling amplitude between them being controlled by the gate voltage on the grain. We find the Josephson current and its low-frequency fluctuations, and predict their periodic dependence with period 2e on the induced charge Q x =CV g .
Spectral density of current fluctuations in a short ballistic superconducting quantum point contact is calculated for arbitrary bias voltages VVV. Contrary to a commonly held view that the supercurrent flow in Josephson junctions is a coherent process without fluctuations, we find extremely large current noise that is caused by the supercurrent coherence. At finite voltages, this noise can be qualitatively understood as the shot noise of the large charge quanta of magnitude $2\ensuremath{\Delta}/V$ equal to the charge transferred through the contact during one period of the Josephson oscillations.
Josephson junctions are the central nonlinear element of superconducting qubits, yet their standard description is reached through a chain of approximations. We propose a purely quantum-mechanical model of the junction, formulated directly in Hilbert space: a charged particle moves on a one-dimensional loop, and the junction is a point at which the wave function obeys self-adjoint boundary conditions, labelled by a coupling strength λλλ, which controls the transparency of the junction, and a phase jump δδδ, which plays the role of the superconducting phase difference. For a loop with a single junction we solve the spectral problem and show that the ground-state momentum expectation, identified with the supercurrent, factorises exactly as ⟨p^0⟩=Ic(δ,λ)sin⁡δ\langle\hat{p}_0\rangle = I_c(δ,λ)\sinδ⟨p^​0​⟩=Ic​(δ,λ)sinδ. For opaque junctions IcI_cIc​ becomes independent of δδδ and the first Josephson relation I=Icsin⁡δI=I_c\sinδI=Ic​sinδ is recovered; for strongly attractive coupling, where the ground state is bound to the junction, the sin⁡δ\sinδsinδ law is also recovered, but with an exponentially suppressed amplitude. For more transparent junctions, however, the model \emph{predicts} deviations from the pure sin⁡δ\sinδsinδ law: IcI_cIc​ acquires a dependence on δδδ already at order λ−2λ^{-2}λ−2, compatible with the higher Josephson harmonics recently found necessary to describe superconducting qubit spectra, and a fully transparent contact yields a current linear in δδδ. When the phase varies in time, a gauge transformation shows that the voltage drop across the junction is proportional to δ˙\dotδδ˙, so that the second Josephson relation emerges without being imposed. The boundary-condition approach thus reproduces both Josephson relations from first principles and predicts corrections to them, offering a route towards higher-fidelity models of superconducting qubits.
We study the Josephson current through a superconducting grain coupled by two tunnel junctions to macroscopic superconducting leads. As a result of the Coulomb blockade we predict the critical Josephson current to have a threshold behavior when the superconducting gap is suppressed below the charging energy by a magnetic field. This effect is a direct consequence of a discrimination by the superconducting ground state between odd and even number of electrons on the grain. The critical current is periodically modulated by the voltage of a gate electrostatically coupled to the grain. We find a characteristic change in this modulation at the threshold.
We measure current across a voltage-biased, long array of Josephson junctions. Current-voltage characteristics do not reveal any critical current and instead display the presence of threshold voltage and finite activation energy, consistent with the Coulomb blockade. However, the respective energy scale is well below the value prescribed by a single-island geometric capacitances. We explain it by strong quantum fluctuations of charge associated with a large value of Josephson energy, EJ≫ECE_J\gg E_CEJ​≫EC​, where ECE_CEC​ is the charging energy associated with the inter-island capacitance. Comparison between the data and theory points to an exponential (in parameter EJ/ECE_J/E_CEJ​/EC​) suppression of the threshold. We achieve a quantitative agreement with theory by accounting for charge-eee solitons associated with the quasiparticles entering the array from its leads. This scenario however underestimates the current below threshold in comparison with the experiment.