In 1998 experiments on InAs/AlSb quantum wells contacted by superconducting Nb electrodes patterned as a grating (1 µm period) of Josephson weak links, the zero-bias differential resistance (the slope of the current–voltage curve at zero current) below about 8 K was found to be thermally activated, with an activation energy that grows with sample width — from roughly 0.7 meV at 16 µm to roughly 23 meV at 200 µm — and is independent of the voltage-probe spacing in co-processed devices. The source paper presented a detailed quantitative study of the temperature dependence of this residual resistance and of its peculiar interrelation with the geometry of the devices, but the microscopic origin of the activated, width-proportional behavior was left open. In the group's companion paper on the same arrays the behavior was found to fall outside the range of established theories and was attributed to a proposed fluctuation effect caused by giant shot noise associated with multiple Andreev reflections, with a characteristic current proportional to device width; the microscopic theory of this proposal was explicitly left undeveloped. The problem is to determine the resistive mechanism at the microscopic level.

Background

When a semiconductor is placed in good electrical contact with a conventional superconductor, superconducting correlations leak into the semiconductor over a coherence length — the proximity effect — so that a short non-superconducting channel joining two superconducting electrodes (a weak link) can carry a supercurrent by the Josephson effect (the dissipationless flow of Cooper pairs, bound electron pairs, across the link). InAs is an attractive semiconductor for such hybrids: a quantum well (a thin semiconductor layer confining electrons to two dimensions, forming a two-dimensional electron gas, 2DEG) between AlSb barriers provides high electron density and mobility, and the InAs surface accumulates electrons, enabling transparent contact to metals. Patterning superconducting Nb electrodes (a conventional superconductor with critical temperature near 9 K) as a fine grating of 1 µm periodicity on the InAs 2DEG produces a long series array of Nb/InAs/Nb Josephson weak links. Such arrays showed flux-periodic resistance oscillations (resistance oscillations with a fixed period in the applied magnetic flux) (M. Thomas et al., Phys. Rev. B 54, R2311 (1996)) and Josephson behavior (H. Drexler et al., Surface Science 361–362, 306 (1996)), establishing them as coherent weak-link arrays.

The residual-resistance phenomenology found in the 1998 source work is the subject of this problem. On cooling below about 8 K, the zero-bias differential resistance (the slope dV/dIdV/dI of the current–voltage curve at zero current) of the grating arrays drops over orders of magnitude but does not reach zero; instead it is thermally activated, meaning it follows R∝exp⁡(−Ea/kBT)R \propto \exp(-E_a/k_B T), with activation energy EaE_a (kBk_B is Boltzmann's constant, TT the temperature). Three facts characterize it: (a) the activated form holds across the measured devices; (b) EaE_a grows with the sample width ww, from roughly 0.7 meV at w≈16w \approx 16 µm to roughly 23 meV at w≈200w \approx 200 µm; (c) in devices from the same process chip, EaE_a depends on width but not on the spacing between voltage probes (about 2.6 meV for w≈65w \approx 65 µm at both 300 µm and 425 µm probe spacings; about 3.6 meV for w≈90w \approx 90 µm at both spacings). Fact (c) rules out explanations in which the resistance is uniformly distributed along the current path.

To account for these observations, the group's open-access companion paper on the same arrays (M. Thomas, H.-R. Blank, K. C. Wong, H. Kroemer, E. L. Hu, Phys. Rev. B 58, 11676 (1998)) noted that the width-proportional activation energy falls outside the range of established theories — a force-fit of the Ambegaokar–Halperin theory of thermally activated fluctuations overshoots the observed activation energy of the 65 µm samples by a factor of about 87 — and proposed that the residual resistance is a fluctuation effect caused by giant shot noise associated with multiple Andreev reflections: with a characteristic (corner) current IcI_c proportional to the device width, the zero-bias resistance takes the activated form R(0)/Ra=exp⁡[−ℏIc/(γekBT)]R(0)/R_a = \exp[-\hbar I_c/(\gamma e k_B T)], where γ\gamma is a numerical constant, ee the elementary charge, and RaR_a a resistance scale, yielding an activation energy proportional to width. The proposal was explicitly qualitative: the authors stated that working out its microscopic details went beyond the scope of that paper, so the mechanism, and the derivation of the activated width scaling, remained open.

First, the Nb/InAs-AlSb material system was largely retired in the 2010s in favor of InAs-based wells coupled to epitaxial thin-film Al, which yields more transparent, uniformly proximitized hybrids. Second, residual dissipation occurs in semiconductor-based Josephson-junction arrays on other platforms — for example, anomalous metallic phases (states that are neither superconducting nor insulating, with a finite, nearly temperature-independent sheet resistance) have been reported in gated epitaxial Al/InAs junction arrays (C. G. L. Bøttcher et al., Phys. Rev. B 110, L180502 (2024)). What is missing, and what this problem asks for, is the microscopic origin of the width-activated resistance in the original system.

Problem Statement

Determine, at the microscopic level, the mechanism that produces the residual zero-bias differential resistance (the slope dV/dIdV/dI of the current–voltage curve at zero current) of superconductor-coupled InAs/AlSb quantum-well grating weak-link arrays below about 8 K. The mechanism must be consistent with the three experimental facts established by the source work: (a) the zero-bias differential resistance is thermally activated, R∝exp⁡(−Ea/kBT)R \propto \exp(-E_a/k_B T), across the measured devices, where TT is the temperature and kBk_B is Boltzmann's constant; (b) the activation energy grows with the sample width ww, from about 0.7 meV at w≈16w \approx 16 µm to about 23 meV at w≈200w \approx 200 µm; (c) for devices co-processed on the same chip, the activation energy depends on width and not on voltage-probe spacing (about 2.6 meV for w≈65w \approx 65 µm at both 300 µm and 425 µm spacings; about 3.6 meV for w≈90w \approx 90 µm at both spacings). Specifically: (i) identify the physical origin of the dissipation; and (ii) derive, or tightly constrain from microscopic physics, the temperature and width dependence of the activation energy, and determine whether the activated form with an activation energy proportional to width actually follows.

References

  1. M. Thomas, H. Kroemer, H.-R. Blank, K. C. Wong, "Induced superconductivity and residual resistance in InAs quantum wells contacted with superconducting Nb electrodes", Physica E: Low-dimensional Systems and Nanostructures 2(1–4), 894–898 (1998). DOI: 10.1016/s1386-9477(98)00182-9, https://doi.org/10.1016/s1386-9477(98)00182-9
  2. H. Kroemer, "Supercurrent flow through a semiconductor: the transport properties of superconductor–semiconductor hybrid structures", Physica E: Low-dimensional Systems and Nanostructures 2(1–4), 887–893 (1998). DOI: 10.1016/s1386-9477(98)00181-7, https://doi.org/10.1016/s1386-9477(98)00181-7
  3. M. Thomas, H.-R. Blank, K. C. Wong, C. Nguyen, H. Kroemer, E. L. Hu, "Flux-periodic resistance oscillations in arrays of superconducting weak links based on InAs-AlSb quantum wells with Nb electrodes", Physical Review B 54, R2311–R2314 (1996). DOI: 10.1103/PhysRevB.54.R2311, https://doi.org/10.1103/PhysRevB.54.R2311
  4. H. Drexler, J. G. E. Harris, E. L. Yuh, K. C. Wong, S. J. Allen, E. G. Gwinn, H. Kroemer, E. L. Hu, "Superconductivity and the Josephson effect in a periodic array of Nb–InAs–Nb junctions", Surface Science 361–362, 306–310 (1996). DOI: 10.1016/0039-6028(96)00409-8, https://doi.org/10.1016/0039-6028(96)00409-8
  5. M. Thomas, H.-R. Blank, K. C. Wong, H. Kroemer, E. L. Hu, "Current-voltage characteristics of semiconductor-coupled superconducting weak links with large electrode separations", Physical Review B 58, 11676 (1998). DOI: 10.1103/PhysRevB.58.11676, arXiv: cond-mat/9806293, https://arxiv.org/abs/cond-mat/9806293
  6. H. Kroemer, E. Hu, "Semiconducting and Superconducting Physics and Devices in the InAs/AlSb Materials System", in G. Timp (ed.), Nanotechnology, Springer, New York, 629–688 (1999). DOI: 10.1007/978-1-4612-0531-9_15, https://doi.org/10.1007/978-1-4612-0531-9_15
  7. T. Rizk, A. Yulius, W. I. Yoo, P. F. Bagwell, D. McInturff, P. Chin, J. M. Woodall, T. M. Pekarek, T. N. Jackson, "Ballistic transport and Andreev resonances in Nb/In superconducting contacts to InAs and LTG-GaAs", Superlattices and Microstructures 25, 757–766 (1999). DOI: 10.1006/spmi.1999.0757, arXiv: cond-mat/9904212, https://arxiv.org/abs/cond-mat/9904212
  8. C. G. L. Bøttcher, F. Nichele, J. Shabani, C. J. Palmstrøm, C. M. Marcus, "Berezinskii–Kosterlitz–Thouless transition and anomalous metallic phase in a hybrid Josephson junction array", Physical Review B 110, L180502 (2024). DOI: 10.1103/PhysRevB.110.L180502, arXiv: 2210.00318, https://arxiv.org/abs/2210.00318