Josephson Junctions

14,000,000 Leading Edge Experts on the ideXlab platform

Scan Science and Technology

Contact Leading Edge Experts & Companies

Scan Science and Technology

Contact Leading Edge Experts & Companies

The Experts below are selected from a list of 20076 Experts worldwide ranked by ideXlab platform

D J Van Harlingen - One of the best experts on this subject based on the ideXlab platform.

  • observation of nonsinusoidal current phase relation in graphene Josephson Junctions
    Physical Review B, 2016
    Co-Authors: Christopher English, D R Hamilton, Cesar Chialvo, I C Moraru, Nadya Mason, D J Van Harlingen
    Abstract:

    The current-phase relation of a Josephson junction can reveal valuable information about the processes influencing the supercurrent. In this paper we present direct measurements of the current-phase relation for Josephson Junctions having a graphene barrier, obtained by a phase-sensitive SQUID interferometry technique. We find that the current-phase relation is forward skewed with respect to the commonly observed sinusoidal behavior for short Junctions in the quasiballistic transport regime, consistent with predictions for the behavior of Dirac fermions in a Josephson junction. The skewness increases with critical current and decreases sharply with increasing temperature.

  • evidence for an anomalous current phase relation in topological insulator Josephson Junctions
    Nature Communications, 2015
    Co-Authors: Cihan Kurter, A D K Finck, Yew San Hor, D J Van Harlingen
    Abstract:

    Josephson Junctions with topological insulator weak links can host low-energy Andreev-bound states giving rise to a current-phase relation that deviates from sinusoidal behaviour. Of particular interest are zero-energy Majorana-bound states that form at a phase difference of π. Here we report on interferometry studies of Josephson Junctions and superconducting quantum interference devices (SQUIDs) incorporating topological insulator weak links. We find that the nodes in single-junction diffraction patterns and SQUID oscillations are lifted and independent of chemical potential. At high temperatures, the SQUID oscillations revert to conventional behaviour, ruling out asymmetry. The node-lifting of the SQUID oscillations is consistent with low-energy Andreev-bound states exhibiting a nonsinusoidal current-phase relation, co-existing with states possessing a conventional sinusoidal current-phase relation. However, the finite nodal currents in the single-junction diffraction pattern suggest an anomalous contribution to the supercurrent possibly carried by Majorana-bound states, although we also consider the possibility of inhomogeneity.

  • low frequency resistance and critical current fluctuations in al based Josephson Junctions
    Applied Physics Letters, 2013
    Co-Authors: Christopher Nugroho, V Orlyanchik, D J Van Harlingen
    Abstract:

    We present low-temperature measurements of the low-frequency 1/f noise arising from an ensemble of two-level fluctuators in the oxide barrier of Al/AlOx/Al Josephson Junctions. The fractional noise power spectrum of the critical-current and normal-state resistance has similar magnitudes and scale linearly with temperature, implying an equivalence between the two. Compiling our results and published data, we deduce the area and temperature scaling of the noise for AlOx barrier Junctions. We find that the density of two-level fluctuators in the junction barrier is similar to the typical value in glassy systems. We discuss the implications and consistency with recent qubit experiments.

  • low frequency resistance and critical current fluctuations in al based Josephson Junctions
    arXiv: Superconductivity, 2013
    Co-Authors: Christopher Nugroho, V Orlyanchik, D J Van Harlingen
    Abstract:

    We present low-temperature measurements of the low-frequency $1/f$ noise arising from an ensemble of two-level fluctuators in the oxide barrier of Al/AlO$_{x}$/Al Josephson Junctions. The fractional noise power spectrum of the critical-current and normal-state resistance have similar magnitudes and scale linearly with temperature, implying an equivalence between the two. Compiling our results and published data, we deduce the area and temperature scaling of the noise for AlO$_{x}$ barrier Junctions. We find that the density of two-level fluctuators in the junction barrier is similar to the typical value in glassy systems. We discuss the implications and consistency with recent qubit experiments.

  • low frequency critical current noise in Josephson Junctions induced by temperature fluctuations
    Applied Physics Letters, 2012
    Co-Authors: S M Anton, Christopher Nugroho, Jeffrey Birenbaum, S R Okelley, V Orlyanchik, Allison Dove, Gustaf Olson, Zack Yoscovits, James N Eckstein, D J Van Harlingen
    Abstract:

    We demonstrate a spurious contribution to low-frequency critical current noise in Josephson Junctions—normally attributed to charge trapping in the barrier—arising from temperature instabilities inherent in cryogenic systems. These temperature fluctuations modify the critical current via its temperature dependence. Cross-correlations between measured temperature and critical current noise in Al-AlOx-Al Junctions show that, despite excellent temperature stability, temperature fluctuations induce observable critical current fluctuations. Particularly, because 1/f critical current noise has decreased with improved fabrication techniques in recent years, it is important to understand and eliminate this additional noise source.

Norman O. Birge - One of the best experts on this subject based on the ideXlab platform.

  • supercurrent in ferromagnetic Josephson Junctions with heavy metal interlayers
    Bulletin of the American Physical Society, 2018
    Co-Authors: Nathan Satchell, Norman O. Birge
    Abstract:

    The length scale over which supercurrent from conventional BCS, s-wave superconductors (S) can penetrate an adjacent ferromagnetic (F) layer depends on the ability to convert singlet Cooper pairs into triplet Cooper pairs. Spin-aligned triplet Cooper pairs are not dephased by the ferromagnetic exchange interaction and can thus penetrate an F layer over much longer distances than singlet Cooper pairs. These triplet Cooper pairs carry a dissipationless spin current and are the fundamental building block for the fledgling field of superspintronics. Singlet-triplet conversion by inhomogeneous magnetism is well established. Here, we describe an attempt to use spin-orbit coupling as an alternative mechanism to mediate singlet-triplet conversion in S-F-S Josephson Junctions. We report that the addition of thin Pt spin-orbit-coupling layers in our Josephson Junctions significantly increases supercurrent transmission, however the decay length of the supercurrent is not found to increase. We attribute the increased supercurrent transmission to Pt acting as a buffer layer to improve the growth of the Co F layer.

  • spin valve Josephson Junctions for cryogenic memory
    arXiv: Superconductivity, 2017
    Co-Authors: W. P. Pratt, Bethany Niedzielski, T J Bertus, Joseph A Glick, R Loloee, Norman O. Birge
    Abstract:

    Josephson Junctions containing two ferromagnetic layers are being considered for use in cryogenic memory. Our group recently demonstrated that the ground-state phase difference across such a junction with carefully chosen layer thicknesses could be controllably toggled between zero and $\pi$ by switching the relative magnetization directions of the two layers between the antiparallel and parallel configurations. However, several technological issues must be addressed before those Junctions can be used in a large-scale memory. Many of these issues can be more easily studied in single Junctions, rather than in the Superconducting QUantum Interference Device (SQUID) used for the phase-sensitive measurements. In this work, we report a comprehensive study of spin-valve Junctions containing a Ni layer with a fixed thickness of 2.0 nm, and a NiFe layer of thickness varying between 1.1 and 1.8 nm in steps of 0.1 nm. We extract the field shift of the Fraunhofer patterns and the critical currents of the Junctions in the parallel and antiparallel magnetic states, as well as the switching fields of both magnetic layers. We also report a partial study of similar Junctions containing a slightly thinner Ni layer of 1.6 nm and the same range of NiFe thicknesses. Unfortunately, current theoretical models of spin-valve Josephson Junctions are not able to describe both data sets with a single set of fit parameters.

  • amplitude control of the spin triplet supercurrent in s f s Josephson Junctions
    Physical Review Letters, 2016
    Co-Authors: William Martinez, W. P. Pratt, Norman O. Birge
    Abstract:

    Josephson Junctions made with conventional s-wave superconductors and containing multiple layers of ferromagnetic materials can carry spin-triplet supercurrent in the presence of certain types of magnetic inhomogeneity. In Junctions containing three ferromagnetic layers, the triplet supercurrent is predicted to be maximal when the magnetizations of the adjacent layers are orthogonal, and zero when the magnetizations of any two adjacent layers are parallel. Here we demonstrate on-off control of the spin-triplet supercurrent in such Junctions, achieved by rotating the magnetization direction of one of the three layers by 90°. We obtain “on-off” ratios of 5, 7, and 19 for the supercurrent in the three samples that have been studied so far. In conclusion, these observations directly confirm one of the most salient predictions of the theory, and they pave the way for applications of spin-triplet Josephson Junctions in the nascent area of “superconducting spintronics”.

  • area dependence of spin triplet supercurrent in ferromagnetic Josephson Junctions
    Journal of Physics: Conference Series, 2012
    Co-Authors: Yixing Wang, W. P. Pratt, Norman O. Birge
    Abstract:

    In 2010, several experimental groups obtained compelling evidence for spin-triplet supercurrent in Josephson Junctions containing strong ferromagnetic materials. Our own best results were obtained from large-area Junctions containing a thick central Co/Ru/Co "synthetic antiferromagnet" and two thin outer layers made of Ni or PdNi alloy. Because the ferromagnetic layers in our samples are multi-domain, one would expect the sign of the local current-phase relation inside the Junctions to vary randomly as a function of lateral position. Here we report measurements of the area dependence of the critical current in several samples, where we find some evidence for those random sign variations. When the samples are magnetized, however, the critical current becomes clearly proportional to the area, indicating that the current-phase relation has the same sign across the entire area of the Junctions.

  • area dependence of spin triplet supercurrent in ferromagnetic Josephson Junctions
    Physical Review B, 2012
    Co-Authors: Yixing Wang, W. P. Pratt, Norman O. Birge
    Abstract:

    Josephson Junctions containing multiple ferromagnetic layers can carry spin-triplet supercurrent under certain conditions. Large-area Junctions containing multiple domains are expected to have a random distribution of 0 or pi coupling across the junction surface, whereas magnetized samples should have uniquely pi coupling everywhere. We have measured the area dependence of the critical current in such Junctions, and confirm that the critical current scales linearly with area in magnetized Junctions. For as-grown (multi-domain) samples, the results are mixed. Samples grown on a thick Nb base exhibit critical currents that scale sub-linearly with area, while samples grown on a smoother Nb/Al multilayer base exhibit critical currents that scale linearly with area. The latter results are consistent with a theoretical picture due to Zyuzin and Spivak that predicts that the as-grown samples should have global pi/2 coupling.

W. P. Pratt - One of the best experts on this subject based on the ideXlab platform.

  • spin valve Josephson Junctions for cryogenic memory
    arXiv: Superconductivity, 2017
    Co-Authors: W. P. Pratt, Bethany Niedzielski, T J Bertus, Joseph A Glick, R Loloee, Norman O. Birge
    Abstract:

    Josephson Junctions containing two ferromagnetic layers are being considered for use in cryogenic memory. Our group recently demonstrated that the ground-state phase difference across such a junction with carefully chosen layer thicknesses could be controllably toggled between zero and $\pi$ by switching the relative magnetization directions of the two layers between the antiparallel and parallel configurations. However, several technological issues must be addressed before those Junctions can be used in a large-scale memory. Many of these issues can be more easily studied in single Junctions, rather than in the Superconducting QUantum Interference Device (SQUID) used for the phase-sensitive measurements. In this work, we report a comprehensive study of spin-valve Junctions containing a Ni layer with a fixed thickness of 2.0 nm, and a NiFe layer of thickness varying between 1.1 and 1.8 nm in steps of 0.1 nm. We extract the field shift of the Fraunhofer patterns and the critical currents of the Junctions in the parallel and antiparallel magnetic states, as well as the switching fields of both magnetic layers. We also report a partial study of similar Junctions containing a slightly thinner Ni layer of 1.6 nm and the same range of NiFe thicknesses. Unfortunately, current theoretical models of spin-valve Josephson Junctions are not able to describe both data sets with a single set of fit parameters.

  • amplitude control of the spin triplet supercurrent in s f s Josephson Junctions
    Physical Review Letters, 2016
    Co-Authors: William Martinez, W. P. Pratt, Norman O. Birge
    Abstract:

    Josephson Junctions made with conventional s-wave superconductors and containing multiple layers of ferromagnetic materials can carry spin-triplet supercurrent in the presence of certain types of magnetic inhomogeneity. In Junctions containing three ferromagnetic layers, the triplet supercurrent is predicted to be maximal when the magnetizations of the adjacent layers are orthogonal, and zero when the magnetizations of any two adjacent layers are parallel. Here we demonstrate on-off control of the spin-triplet supercurrent in such Junctions, achieved by rotating the magnetization direction of one of the three layers by 90°. We obtain “on-off” ratios of 5, 7, and 19 for the supercurrent in the three samples that have been studied so far. In conclusion, these observations directly confirm one of the most salient predictions of the theory, and they pave the way for applications of spin-triplet Josephson Junctions in the nascent area of “superconducting spintronics”.

  • area dependence of spin triplet supercurrent in ferromagnetic Josephson Junctions
    Journal of Physics: Conference Series, 2012
    Co-Authors: Yixing Wang, W. P. Pratt, Norman O. Birge
    Abstract:

    In 2010, several experimental groups obtained compelling evidence for spin-triplet supercurrent in Josephson Junctions containing strong ferromagnetic materials. Our own best results were obtained from large-area Junctions containing a thick central Co/Ru/Co "synthetic antiferromagnet" and two thin outer layers made of Ni or PdNi alloy. Because the ferromagnetic layers in our samples are multi-domain, one would expect the sign of the local current-phase relation inside the Junctions to vary randomly as a function of lateral position. Here we report measurements of the area dependence of the critical current in several samples, where we find some evidence for those random sign variations. When the samples are magnetized, however, the critical current becomes clearly proportional to the area, indicating that the current-phase relation has the same sign across the entire area of the Junctions.

  • area dependence of spin triplet supercurrent in ferromagnetic Josephson Junctions
    Physical Review B, 2012
    Co-Authors: Yixing Wang, W. P. Pratt, Norman O. Birge
    Abstract:

    Josephson Junctions containing multiple ferromagnetic layers can carry spin-triplet supercurrent under certain conditions. Large-area Junctions containing multiple domains are expected to have a random distribution of 0 or pi coupling across the junction surface, whereas magnetized samples should have uniquely pi coupling everywhere. We have measured the area dependence of the critical current in such Junctions, and confirm that the critical current scales linearly with area in magnetized Junctions. For as-grown (multi-domain) samples, the results are mixed. Samples grown on a thick Nb base exhibit critical currents that scale sub-linearly with area, while samples grown on a smoother Nb/Al multilayer base exhibit critical currents that scale linearly with area. The latter results are consistent with a theoretical picture due to Zyuzin and Spivak that predicts that the as-grown samples should have global pi/2 coupling.

  • optimization of spin triplet supercurrent in ferromagnetic Josephson Junctions
    Physical Review Letters, 2012
    Co-Authors: Carolin Klose, T P Ginley, Trupti Khaire, Norman O. Birge, W. P. Pratt, J A Borchers, Benjamin J Mcmorran, Brian J. Kirby, Yixing Wang, Brian B Maranville
    Abstract:

    We have observed long-range spin-triplet supercurrents in Josephson Junctions containing ferromagnetic (F) materials, which are generated by noncollinear magnetizations between a central Co=Ru=Co synthetic antiferromagnet and two outer thin F layers. Here we show that the spin-triplet supercurrent is enhanced up to 20 times after our samples are subject to a large in-plane field. This occurs because the synthetic antiferromagnet undergoes a ‘‘spin-flop’’ transition, whereby the two Co layer magnetizations end up nearly perpendicular to the magnetizations of the two thin F layers. We report direct experimental evidence for the spin-flop transition from scanning electron microscopy with polarization analysis and from spin-polarized neutron reflectometry. These results represent a first step toward experimental control of spin-triplet supercurrents.

Kenji Watanabe - One of the best experts on this subject based on the ideXlab platform.

  • anomalous phase dynamics of driven graphene Josephson Junctions
    Physical Review Research, 2020
    Co-Authors: Sandesh S Kalantre, Kenji Watanabe, Takashi Taniguchi, Francois Amet, Mingtso Wei, M Hernandezrivera, J R Williams
    Abstract:

    This paper draws a bridge between non-linear dynamics and the study of exotic topological phases in quantum materials with Josephson Junctions. The authors probe a graphene junction in the underdamped regime and show the consequences of nonlinear and chaotic phase dynamics on Shapiro measurements.

  • anomalous phase dynamics of driven graphene Josephson Junctions
    arXiv: Mesoscale and Nanoscale Physics, 2019
    Co-Authors: Sandesh S Kalantre, Kenji Watanabe, Takashi Taniguchi, Francois Amet, Mingtso Wei, M Hernandezrivera, J R Williams
    Abstract:

    Josephson Junctions with weak-links of exotic materials allow the elucidation of the Josephson effect in previously unexplored regimes. Further, such devices offer a direct probe of novel material properties, for example in the search for Majorana fermions. In this work, we report on DC and AC Josephson effect of high-mobility, hexagonal boron nitride (h-BN) encapsulated graphene Josephson Junctions. On the application of RF radiation, we measure phase-locked Shapiro steps. An unexpected bistability between $\pm 1$ steps is observed with switching times on the order of seconds. A critical scaling of a bistable state is measured directly from the switching time, allowing for direct comparison to numerical simulations. We show such intermittent chaotic behavior is a consequence of the nonlinear dynamics of the junction and has a sensitive dependence on the current-phase relation. This work draws connections between nonlinear phenomena in dynamical systems and their implications for ongoing condensed matter experiments exploring topology and exotic physics.

  • supercurrent flow in multiterminal graphene Josephson Junctions
    Nano Letters, 2019
    Co-Authors: Anne Draelos, Andrew Seredinski, Hengming Li, Y Mehta, Kenji Watanabe, Takashi Taniguchi, Ivan Borzenets, Francois Amet, Gleb Finkelstein
    Abstract:

    We investigate the electronic properties of ballistic planar Josephson Junctions with multiple superconducting terminals. Our devices consist of monolayer graphene encapsulated in boron nitride with molybdenum–rhenium contacts. Resistance measurements yield multiple resonant features, which are attributed to supercurrent flow among adjacent and nonadjacent Josephson Junctions. In particular, we find that superconducting and dissipative currents coexist within the same region of graphene. We show that the presence of dissipative currents primarily results in electron heating and estimate the associated temperature rise. We find that the electrons in encapsulated graphene are efficiently cooled through the electron–phonon coupling.

  • magnetic field compatible circuit quantum electrodynamics with graphene Josephson Junctions
    Nature Communications, 2018
    Co-Authors: James Kroll, Kenji Watanabe, Takashi Taniguchi, Srijit Goswami, Willemijn Uilhoorn, K L Van Der Enden, D De Jong, Maja C Cassidy, L P Kouwenhoven
    Abstract:

    Circuit quantum electrodynamics has proven to be a powerful tool to probe mesoscopic effects in hybrid systems and is used in several quantum computing (QC) proposals that require a transmon qubit able to operate in strong magnetic fields. To address this we integrate monolayer graphene Josephson Junctions into microwave frequency superconducting circuits to create graphene based transmons. Using dispersive microwave spectroscopy we resolve graphene's characteristic band dispersion and observe coherent electronic interference effects confirming the ballistic nature of our graphene Josephson Junctions. We show that the monoatomic thickness of graphene renders the device insensitive to an applied magnetic field, allowing us to perform energy level spectroscopy of the circuit in a parallel magnetic field of 1 T, an order of magnitude higher than previous studies. These results establish graphene based superconducting circuits as a promising platform for QC and the study of mesoscopic quantum effects that appear in strong magnetic fields.

  • ballistic graphene Josephson Junctions from the short to the long junction regimes
    Physical Review Letters, 2016
    Co-Authors: Ivan Borzenets, Anne Draelos, Andrew Seredinski, Kenji Watanabe, Takashi Taniguchi, Francois Amet, Mingtso Wei, Yuriy Bomze, Michihisa Yamamoto, S Tarucha
    Abstract:

    We investigate the critical current I_{C} of ballistic Josephson Junctions made of encapsulated graphene-boron-nitride heterostructures. We observe a crossover from the short to the long junction regimes as the length of the device increases. In long ballistic Junctions, I_{C} is found to scale as ∝exp(-k_{B}T/δE). The extracted energies δE are independent of the carrier density and proportional to the level spacing of the ballistic cavity. As T→0 the critical current of a long (or short) junction saturates at a level determined by the product of δE (or Δ) and the number of the junction's transversal modes.

Takashi Taniguchi - One of the best experts on this subject based on the ideXlab platform.

  • anomalous phase dynamics of driven graphene Josephson Junctions
    Physical Review Research, 2020
    Co-Authors: Sandesh S Kalantre, Kenji Watanabe, Takashi Taniguchi, Francois Amet, Mingtso Wei, M Hernandezrivera, J R Williams
    Abstract:

    This paper draws a bridge between non-linear dynamics and the study of exotic topological phases in quantum materials with Josephson Junctions. The authors probe a graphene junction in the underdamped regime and show the consequences of nonlinear and chaotic phase dynamics on Shapiro measurements.

  • anomalous phase dynamics of driven graphene Josephson Junctions
    arXiv: Mesoscale and Nanoscale Physics, 2019
    Co-Authors: Sandesh S Kalantre, Kenji Watanabe, Takashi Taniguchi, Francois Amet, Mingtso Wei, M Hernandezrivera, J R Williams
    Abstract:

    Josephson Junctions with weak-links of exotic materials allow the elucidation of the Josephson effect in previously unexplored regimes. Further, such devices offer a direct probe of novel material properties, for example in the search for Majorana fermions. In this work, we report on DC and AC Josephson effect of high-mobility, hexagonal boron nitride (h-BN) encapsulated graphene Josephson Junctions. On the application of RF radiation, we measure phase-locked Shapiro steps. An unexpected bistability between $\pm 1$ steps is observed with switching times on the order of seconds. A critical scaling of a bistable state is measured directly from the switching time, allowing for direct comparison to numerical simulations. We show such intermittent chaotic behavior is a consequence of the nonlinear dynamics of the junction and has a sensitive dependence on the current-phase relation. This work draws connections between nonlinear phenomena in dynamical systems and their implications for ongoing condensed matter experiments exploring topology and exotic physics.

  • supercurrent flow in multiterminal graphene Josephson Junctions
    Nano Letters, 2019
    Co-Authors: Anne Draelos, Andrew Seredinski, Hengming Li, Y Mehta, Kenji Watanabe, Takashi Taniguchi, Ivan Borzenets, Francois Amet, Gleb Finkelstein
    Abstract:

    We investigate the electronic properties of ballistic planar Josephson Junctions with multiple superconducting terminals. Our devices consist of monolayer graphene encapsulated in boron nitride with molybdenum–rhenium contacts. Resistance measurements yield multiple resonant features, which are attributed to supercurrent flow among adjacent and nonadjacent Josephson Junctions. In particular, we find that superconducting and dissipative currents coexist within the same region of graphene. We show that the presence of dissipative currents primarily results in electron heating and estimate the associated temperature rise. We find that the electrons in encapsulated graphene are efficiently cooled through the electron–phonon coupling.

  • magnetic field compatible circuit quantum electrodynamics with graphene Josephson Junctions
    Nature Communications, 2018
    Co-Authors: James Kroll, Kenji Watanabe, Takashi Taniguchi, Srijit Goswami, Willemijn Uilhoorn, K L Van Der Enden, D De Jong, Maja C Cassidy, L P Kouwenhoven
    Abstract:

    Circuit quantum electrodynamics has proven to be a powerful tool to probe mesoscopic effects in hybrid systems and is used in several quantum computing (QC) proposals that require a transmon qubit able to operate in strong magnetic fields. To address this we integrate monolayer graphene Josephson Junctions into microwave frequency superconducting circuits to create graphene based transmons. Using dispersive microwave spectroscopy we resolve graphene's characteristic band dispersion and observe coherent electronic interference effects confirming the ballistic nature of our graphene Josephson Junctions. We show that the monoatomic thickness of graphene renders the device insensitive to an applied magnetic field, allowing us to perform energy level spectroscopy of the circuit in a parallel magnetic field of 1 T, an order of magnitude higher than previous studies. These results establish graphene based superconducting circuits as a promising platform for QC and the study of mesoscopic quantum effects that appear in strong magnetic fields.

  • ballistic graphene Josephson Junctions from the short to the long junction regimes
    Physical Review Letters, 2016
    Co-Authors: Ivan Borzenets, Anne Draelos, Andrew Seredinski, Kenji Watanabe, Takashi Taniguchi, Francois Amet, Mingtso Wei, Yuriy Bomze, Michihisa Yamamoto, S Tarucha
    Abstract:

    We investigate the critical current I_{C} of ballistic Josephson Junctions made of encapsulated graphene-boron-nitride heterostructures. We observe a crossover from the short to the long junction regimes as the length of the device increases. In long ballistic Junctions, I_{C} is found to scale as ∝exp(-k_{B}T/δE). The extracted energies δE are independent of the carrier density and proportional to the level spacing of the ballistic cavity. As T→0 the critical current of a long (or short) junction saturates at a level determined by the product of δE (or Δ) and the number of the junction's transversal modes.