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Physical review. B, Condensed matter

Cooper-pair current through ultrasmall Josephson junctions

Gert‐Ludwig Ingold, Hermann Grabert, Udo Eberhardt

Published Jul 1, 1994• 139 citations
Publisher

Abstract

The current-voltage characteristics of ultrasmall Josephson junctions sensitively depend on the electromagnetic environment of the junction. When the charging energy exceeds the Josephson coupling energy, the usual supercurrent at zero voltage is completely suppressed. However, for typical environmental impedances, which are small compared to the resistance quantum, stochastic Cooper-pair tunneling leads to a supercurrent peak at a small finite voltage which is proportional to the temperature and the low-frequency resistance of the external circuit. An analytic expression for the form of this universal peak, which is independent of the high-frequency behavior of the environment, is given. With increasing Josephson coupling the peak merges into the usual supercurrent of a Josephson junction. At larger voltages the Cooper-pair current depends on details of the environment. Current peaks are shown to result from resonances in the environmental impedance. Specifically, the case of an LC transmission line of finite length is discussed.

DOI
10.1103/physrevb.50.395

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Physical review. B, Condensed matter•Nov 1, 1994•9 citations
The current-voltage characteristics of ultrasmall Josephson junctions sensitively depend on the electromagnetic environment of the junction. When the charging energy exceeds the Josephson coupling energy, the usual supercurrent at zero voltage is completely suppressed. However, for typical environmental impedances, which are small compared to the resistance quantum, stochastic Cooper-pair tunneling leads to a supercurrent peak at a small finite voltage which is proportional to the temperature and the low-frequency resistance of the external circuit. An analytic expression for the form of this universal peak, which is independent of the high-frequency behavior of the environment, is given. With increasing Josephson coupling the peak merges into the usual supercurrent of a Josephson junction. At larger voltages the Cooper-pair current depends on details of the environment. Current peaks are shown to result from resonances in the environmental impedance. Specifically, the case of an LC transmission line of finite length is discussed.
The supercurrent of a Josephson junction is reduced by phase diffusion. For ultrasmall capacitance junctions the current may be further decreased by Coulomb blockade effects. We calculate the Cooper pair current by means of time-dependent perturbation theory to all orders in the Josephson coupling energy and obtain the current-voltage characteristic in closed form in a range of parameters of experimental interest. The results comprehend phase diffusion of the coherent Josephson current in the classical regime as well as the supercurrent peak due to incoherent Cooper pair tunneling in the Coulomb blockade regime.
The Cooper-pair tunneling in superconducting charge boxes in tunable electromagnetic environments is experimentally studied using a single-electron-transistor electrometer and via current-voltage characteristics measurement with small excitations. In the low-impedance regime, the environmental effect is quantified with an effective Josephson coupling energy, which is found to decrease as the environment impedance increases. In the high-impedance regime, the Cooper-pair tunneling is blockaded. For impedance above $250\phantom{\rule{0.3em}{0ex}}\mathrm{M}\ensuremath{\Omega}$, Coulomb oscillation of the zero-bias conductance suggests that Cooper-pair tunneling is completely suppressed and subgap quasiparticle tunneling becomes dominant.
We present observations of charging effects for Cooper pairs in short linear arrays of small-capacitance Josephson junctions. Current-voltage characteristics show a Coulomb gap for Cooper-pair tunneling when the charging energy exceeds the Josephson coupling energy. In a double junction a zero-voltage current is observed that is modulated by a gate voltage applied to the metal island between the junctions. For longer arrays a crossover from Coulomb blockade of Cooper-pair tunneling to a supercurrent is observed when the ratio of Josephson coupling to charging energy is increased.
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.
This thesis presents the results of an experimental study on single-charge effects in nanoscale Josephson junctions and Cooper pair transistors (CPTs). In nanoscale Josephson junctions the charging energy EC becomes significant at sub-Kelvin temperatures and single-charge effects, such as the Coulomb blockade of Cooper pair tunneling, influence the transport properties. In order to observe charging effects in a single Josephson junction, the impedance of the electromagnetic environment surrounding the junction has to be larger than the quantum resistance (RQ=h/4e2≈6.45kΩ). In this work the high impedance environment is obtained by biasing the sample under test (single Josephson junction or CPT) with four one-dimensional Josephson junction arrays having SQUID geometry. The advantage of this configuration is the possibility of tuning in situ the effective impedance of the electromagnetic environment. By applying a magnetic field perpendicular to the SQUID loops, the Josephson energy EJ of the SQUIDs is suppressed, resulting in an increase of the measured zero bias resistance of the arrays of several orders of magnitude (104< R0 (Ω) <109). This bias method enables the measurement of the same sample in environments with different impedance. As the impedance of the environment is increased, the current-voltage characteristics (IVCs) of the single Josephson junction and of the CPT show a well defined Coulomb blockade feature with a region of negative differential resistance, signature of the coherent tunneling of single Cooper pairs. The measured IVCs of a single Josephson junction with SQUID geometry in the high impedance environment show a qualitative agreement with the Bloch band theory as the EJ/EC ratio of the junction is tuned with the magnetic field. We also studied a single nontunable Josephson junction with strong coupling (EJ/EC > 1), where the exact dual of the overdamped Josephson effect is realized, resulting in a dual shape of the IVC, where the roles of current and voltage are exchanged. Here, we make for the first time a detailed quantitative comparison with a theory which includes the effect of fluctuations due to the finite temperature of the environment. The measurements on CPTs in the high impedance environment showed that the Coulomb blockade voltage is modulated periodically by the gate-induced charge. The gate-voltage dependence of the CPT changes from e-periodic to 2e-periodic as the impedance of the environment is increased. The high impedance environment reduces quasiparticle tunneling rates, thereby restoring the even parity of the CPT island. This behavior suggests that high impedance leads can be used to effectively suppress quasiparticle poisoning.
The number of Cooper pairs on a small superconducting island coupled to superconducting leads by a Josephson junction fluctuates due to tunneling of Cooper pairs through the junction. On the other hand, the impedance of the circuit surrounding the island tends to suppress these fluctuations and to restore charge quantization. We consider the influence of the electromagnetic environment within a two-state approximation for the charge on the island. The average charge and the slope of the steps of the Coulomb staircase are calculated for a realistic environmental impedance and finite temperatures in the range of current experimental studies. It is shown that even a low-frequency environment may lead to a considerable reduction of charge fluctuations provided the integrated total impedance is large enough.