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The Josephson effect in nanoscale tunnel junctions P. Joyez, D. Vion, M. Götz*, M. Devoret and D. EsteveService de Physique de l’Etat CondenséCEA-Saclay, France*present address: PTB Braunschweig, GermanyIn nanoscale Josephson junctions, the Josephson coupling energy is usually comparable withthe charging energy of the junction and with the typical energy of thermal fluctuations. Underthese circumstances, phase fluctuations imposed by the electromagnetic environment of thejunction crucially affect the junction electrical behavior. In particular they determine themaximum "supercurrent" the junction can sustain. We discuss this quantity in the case wherethe junction is not resistively shunted, so that the I-V characteristics of the junction remainshysteretic. For a simple, yet realistic unshunted junction model, we obtain detailed predictionsof the shape of the supercurrent branch of the I-V characteristic. Finally we presentexperimental results supporting the theoretical analysis and which demonstrate that thesupercurrent in an unshunted nanoscale Josephson junction can indeed be of the order of itscritical current.Key Words: Josephson effect, supercurrent, phase diffusion, switching current, escape rateRunning title: The Josephson effect in nanoscale tunnel junctions1. IntroductionThe prediction in 1962 of the Josephson effect [1] — and its observation soon afterwards —surprised the specialists of superconductivity at that time. It was found that a supercurrent ofunexpectedly large magnitude can flow between two superconductors separated by an insulatingtunnel barrier. The Josephson effect is a macroscopic quantum phenomenon: the supercurrentresults from the coherent tunneling of Cooper pairs driven by the phase difference d between thecondensates of the two superconductors. Unlike the phase difference that can exist between twopositions of a single particle wavefunction, d is a collective variable directly coupled to macroscopicelectric quantities in the circuit to which the junction is connected. Josephson showed that(1)j0d dþþþþþþþþþþdt=vi=I0 sin dwhere v and i are the voltage and current operators for the junction, I0 is the so-called critical currentof the junction and j0= ¶ 2 e the reduced flux quantum. In large tunnel junctions like those studied immediately after the discovery of the effect, the phaseJournal of Superconductivity, Vol. 12 No. 6, 1999, pp. 757-766.difference behaves as a classical quantity with little thermal fluctuations. The reason is thatrelatively large capacitance of the junction, tends to make the instantaneous voltage v small and thustends to suppress both thermal and quantum fluctuations of d. However, in recent years, it has been possible with the advent of electron beam nanolithography tomake junctions with area – and hence capacitance – so small that the fluctuations of d are no longerdetermined almost essentially by the junction itself but by the circuit in which it is embedded, i.e. itselectromagnetic environment. This can be apprehended by calculating the the r.m.s. amplitude ofphase fluctuations in an approximation that replaces sind by d in (1): For small phase amplitudearound d=0mod2 p the junctions behaves as a linear inductor with inductance L0=j0I0 and inthis case, the r.m.s. amplitude of phase fluctuations are given by [2](2)"#########Xd2\=ikjjjÅ-+ ReZtHwLþþþþþþþþþþþþþþþþþþþþþþþþþRQ 1þþþþþþþþþþþþþþþþþþþþþþþþþþþþ1+e-¶wþþþþþþþþþþþkB T dwþþþþþþþþþþþwy{zzz12where RQ is the resistance quantum h4 e2>6.5kW, kB the Boltzmann constant, T the temperature,and Zt is the total effective impedance of the inductance L0 in parallel with the capacitance of thejunction and with the impedance of the external circuit connected accross the junction. If thejunction is small, its capacitance and effective inductance have high impedance on a broadfrequency range over which Zt is entirely determined by the external circuit connected to thejunction. One sees that depending on the circuit parameters, the fluctuations of d can be large (i.e.comparable or larger than 2p), in which case the instantaneous value of the supercurrentI=Xi\=I0 Xsind\ flowing through the junction is washed out, even at T=0. In order for theinstantaneous supercurrent in a small Josephson junction to reach I0 it is necessary to have smallquantum spreading of the phase. This classical phase behavior requires that the total impedance is abroad-band low impedance HZt`RQL. It turns out that this requirement is easily met, since ordinaryleads connected to a junction are similar to transmission lines and present an impedance of the orderof the vacuum impedance Z0>377W. In fact, having quantum phase fluctuations survive in a smalljunction requires an engineering effort on the environment of the junction; it can be achieved forinstance by microfabricating resistances in the junction leads, close to the junction [3]. As we have just explained, although a classical phase is a necessary condition to have a largesupercurrent, it is however not sufficient to observe a large static supercurrent: the supercurrent maystill classically time-average to nearly zero due to phase diffusion. The simplest way to limit phasediffusion is to shunt the junction on-chip with a low value resistance, and it indeed enables to reachstatic supercurrents of order I0. However, this solution has a number of drawbacks when measuringthe I-V characteristic of the junction : i) the current flowing through the junction and the resistor cannot be measured independently,making it difficult to determine precisely the junction parameters. For instance, the verification ofthe quality of a tunnel junction by the measurement of the subgap quasiparticle current cannot beperformed.ii) the voltage scale of the characteristic is small; it is imposed by the resistor, not by thesuperconducting gap. Measuring it requires a very sensitive voltmeter. iii) the characteristic is non-hysteretic; the shunted junction behaves as a mere non-linear resistor.2On the contrary, the switching behavior of an hysteretic junction provides an easy way to measurethe maximum supercurrent.The purpose of this article