The 1992-1993 experiments of Nguyen, Kroemer, Hu and co-workers on superconducting niobium electrodes deposited directly onto high-density, high-mobility InAs-AlSb quantum wells revealed an anomalous differential-conductance lineshape at 4.2-9 K: a sharp zero-bias peak attributable to enhanced Andreev reflection, accompanied by a broad conductance suppression (dip) at bias voltages several to roughly ten times the superconducting gap voltage, whose position follows the temperature dependence of the niobium gap and which vanishes above the niobium transition temperature (9.2 K). Standard scattering theory of normal-metal-superconductor interfaces predicts conductance structure only at or below the gap voltage, so the above-gap dip lacked a microscopic quantitative explanation. The authors deferred a detailed analysis of multiple normal reflections of electrons and Andreev-reflected holes at the quantum-well interfaces to future work. The group's own 1994-1997 follow-ups established the qualitative mechanism (backplane-enhanced multiple Andreev reflection) and characterized the Josephson-coupling length and temperature scales, but no published work quantitatively derives the bias scale and depth of the above-gap dip.

Background

A quantum well is a thin semiconductor layer that confines electrons to two dimensions; in an InAs-AlSb quantum well a roughly 15 nm indium arsenide (InAs) layer is sandwiched between aluminum antimonide (AlSb) barriers, and the conduction-band offsets trap a two-dimensional electron gas (2DEG). Grown by molecular beam epitaxy with tellurium delta-doping (a planar sheet of dopants placed inside the barriers, spatially separated from the well so that ionized-donor scattering is avoided), these wells reached sheet densities near 8×1012 cm−28\times10^{12}\ \mathrm{cm^{-2}} occupying two size-quantized subbands, mobilities near 105 cm2/Vs10^{5}\ \mathrm{cm^2/Vs} at 4.2 K, and an elastic mean free path on the order of 3 µm. The InAs surface pins its Fermi level above the conduction-band edge, so metallic contacts to InAs are essentially free of Schottky barriers (the rectifying depletion layer that ordinarily forms at a metal-semiconductor interface). This makes an InAs channel a natural weak link between superconducting electrodes: niobium (Nb), with superconducting transition temperature Tc=9.2 KT_c = 9.2\ \mathrm{K} and an energy gap Δ≈1.5 meV\Delta \approx 1.5\ \mathrm{meV}, deposited directly onto the quantum well.

Andreev reflection is the process by which an electron incident on a superconductor with sub-gap energy is retro-reflected as a hole while a Cooper pair enters the condensate; it doubles the charge transferred per injection event and therefore enhances the sub-gap conductance of a transparent contact. The standard Blonder-Tinkham-Klapwijk (BTK) model treats a single normal-metal-superconductor interface with a delta-function barrier of dimensionless strength ZZ. In a two-terminal superconductor-normal-superconductor device, quasiparticles can undergo multiple Andreev reflections (MAR), gaining energy eVeV per round trip between electrodes held at a potential difference VV; the resulting series of conductance features at bias voltages V=2Δ/(ne)V = 2\Delta/(ne), where nn is a positive integer, is called the subharmonic gap structure.

In 1990-1993 the group of Nguyen, Kroemer and Hu studied Nb-(InAs-AlSb quantum well)-Nb devices with 1 µm electrode spacing and measured the differential conductance G(V,T)G(V,T) along the well at 4.2-9 K. Normalized to its normal-state value, GG showed (i) a sharp zero-bias peak - a signature of enhanced Andreev reflection - and, anomalously, (ii) a broad dip, a suppression below the normal-state conductance, at bias voltages several to roughly ten times the superconducting gap voltage (the primary account, Phys. Rev. Lett. 69, 2847 (1992), describes the drop as occurring at voltages an order of magnitude above the gap voltage, with a return to the normal conductance value at yet higher bias). The dip position followed the temperature dependence of the Nb gap, and both features vanished above TcT_c. In addition, no Josephson supercurrent (a dissipationless current supported by the phase difference between two superconductors) was observed for electrode separations above 0.7 µm in the explored range above 4.2 K. Within textbook scattering theory the dip is anomalous: BTK/MAR physics produces structure at or below 2Δ/e2\Delta/e, whereas the observed suppression sits far above the gap yet still tracks Δ(T)\Delta(T). The authors interpreted the anomalies as Andreev reflections modified by multiple normal reflections of both electrons and Andreev-reflected holes between the Nb electrodes and the well's back interface (the bottom AlSb barrier of the well, acting as a backplane that returns particles toward the contacts), in the presence of a nonuniform electrostatic potential along the well, and they explicitly reserved for future work the detailed theoretical and experimental analysis - including the proposal that Andreev-reflected particles are recaptured inside the superconducting electrodes - of how heterointerface reflection coefficients, subband structure, finite mean free path, and Nb-InAs contact properties combine to set the observed bias and temperature scales. Their subsequent papers (1994-1997) established the backplane-reflection enhancement of Andreev reflection experimentally and characterized the length and temperature scales of induced superconductivity in these devices, but the quantitative microscopic description of the above-gap dip was not delivered.

Problem Statement

For Nb electrodes in direct, barrier-free contact with high-density, high-mobility InAs-AlSb quantum wells (sheet density about 8×1012 cm−28\times10^{12}\ \mathrm{cm^{-2}} occupying two subbands, mobility near 105 cm2/Vs10^{5}\ \mathrm{cm^2/Vs}, mean free path about 3 µm; Nb with superconducting transition temperature Tc=9.2 KT_c = 9.2\ \mathrm{K} and gap Δ≈1.5 meV\Delta \approx 1.5\ \mathrm{meV}), establish a microscopic, quantitative account of the superconductivity-induced anomalies of the differential conductance G(V,T)G(V,T) measured along the well at 4.2-9 K: (i) the broad conductance dip at bias voltages several to roughly ten times the superconducting gap voltage, with recovery to the normal-state value at still higher bias; and (ii) the conditions under which enhanced Andreev reflection produces the zero-bias conductance peak together with this higher-bias suppression - specifically, how the heterointerface reflection coefficients, subband structure, finite mean free path, and Nb-InAs contact transparency combine, and what role, if any, is played by multiple normal reflections of electrons and Andreev-reflected holes between the Nb electrodes and the well's bottom barrier in the presence of the nonuniform potential along the well. The account establishes (i) the quantitative bias position of the dip together with the qualitative occurrence of the suppression below the normal-state conductance, and (ii) the conditions under which the zero-bias conductance peak appears; the role played by multiple normal reflections is an exploratory part of (ii) whose answer may be negative, and the ingredient list above describes the space the account covers, not four separate derivations. The account must derive, rather than fit, this reported lineshape from a microscopic model of these devices at the stated parameters: the bias voltage of the conductance minimum and the suppression below the normal-state conductance must be outputs of the model, while the interface quantities it introduces (heterointerface reflection coefficients and Nb-InAs contact transparency) must be computed from the stated materials and device geometry or taken from independent measurements, and must not be adjusted to reproduce the measured G(V,T)G(V,T). The derived conductance minimum must fall within the reported range of roughly several to ten times the superconducting gap voltage and must track the temperature dependence of the Nb gap across the stated 4.2-9 K measurement range rather than at a single temperature; the accompanying suppression below the normal-state conductance is a qualitative consequence of the model.

References

  1. Chanh Nguyen, Herbert Kroemer, Evelyn L. Hu, John H. English, "Low-temperature (4.2-9K) transport along InAs-AlSb quantum wells with delta-doped barriers and superconducting niobium electrodes", Journal of Crystal Growth 127, 845-848 (1993). DOI 10.1016/0022-0248(93)90745-i, https://doi.org/10.1016/0022-0248(93)90745-i
  2. Chanh Nguyen, Herbert Kroemer, Evelyn L. Hu, "Anomalous Andreev conductance in InAs-AlSb quantum well structures with Nb electrodes", Physical Review Letters 69, 2847-2850 (1992). DOI 10.1103/PhysRevLett.69.2847, https://doi.org/10.1103/PhysRevLett.69.2847
  3. Chanh Nguyen, James Werking, Herbert Kroemer, Evelyn L. Hu, "InAs-AlSb quantum well as superconducting weak link with high critical current density", Applied Physics Letters 57, 87-89 (1990). DOI 10.1063/1.103546, https://doi.org/10.1063/1.103546
  4. Herbert Kroemer, Chanh Nguyen, Evelyn L. Hu, Esther L. Yuh, Mason Thomas, Ki C. Wong, "Quasiparticle transport and induced superconductivity in InAs-AlSb quantum wells with Nb electrodes", Physica B: Condensed Matter 203, 298-306 (1994). DOI 10.1016/0921-4526(94)90073-6, https://doi.org/10.1016/0921-4526(94)90073-6
  5. Chanh Nguyen, Herbert Kroemer, Evelyn L. Hu, "Contact resistance of superconductor-semiconductor interfaces: The case of Nb-InAs/AlSb quantum-well structures", Applied Physics Letters 65, 103-105 (1994). DOI 10.1063/1.113047, https://doi.org/10.1063/1.113047
  6. Herbert Kroemer, Chanh Nguyen, Evelyn L. Hu, "Electronic interactions at superconductor-semiconductor interfaces", Solid-State Electronics 37, 1021-1025 (1994). DOI 10.1016/0038-1101(94)90349-2, https://doi.org/10.1016/0038-1101(94)90349-2
  7. Mason Thomas, Hans-Richard Blank, K. C. Wong, Chanh Nguyen, Herbert Kroemer, Evelyn 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
  8. Herbert Kroemer, Mason Thomas, "Induced superconductivity in InAs quantum wells with superconducting contacts", Superlattices and Microstructures 21, 61-67 (1997). DOI 10.1006/spmi.1996.0169, https://doi.org/10.1006/spmi.1996.0169
  9. Mason Thomas, Herbert Kroemer, Hans-Richard Blank, Ki C. Wong, "Induced superconductivity and residual resistance in InAs quantum wells contacted with superconducting Nb electrodes", Physica E: Low-dimensional Systems and Nanostructures 2, 894-898 (1998). DOI 10.1016/S1386-9477(98)00182-9, https://doi.org/10.1016/S1386-9477(98)00182-9
  10. Niko van der Post, Junsaku Nitta, Hideaki Takayanagi, "Elastic scattering and the current-voltage characteristics of superconducting Nb-InAs-Nb junctions", Applied Physics Letters 63, 2555-2557 (1993). DOI 10.1063/1.110431, https://doi.org/10.1063/1.110431
  11. A. Jacobs, R. Kümmel, H. Plehn, "Proximity effect, Andreev reflections, and charge transport in mesoscopic superconducting/semiconducting heterostructures", Superlattices and Microstructures 25, 669-681 (1999). DOI 10.1006/spmi.1999.0718, https://doi.org/10.1006/spmi.1999.0718
  12. A. Vaknin, Z. Ovadyahu, "The above-gap structure in normal-metal - superconductor junctions", Journal of Physics: Condensed Matter 9, L303-L306 (1997). DOI 10.1088/0953-8984/9/19/004, https://doi.org/10.1088/0953-8984/9/19/004
  13. G. Bellomi, W. J. Mitchell, Evelyn L. Hu, "In situ preparation of the semiconductor-superconductor interface: Nb contacts on InAs quantum wells", Journal of Vacuum Science & Technology B 17, 2001-2004 (1999). DOI 10.1116/1.590861, https://doi.org/10.1116/1.590861
  14. T. A. Eckhause, S. Tsujino, K. W. Lehnert, E. G. Gwinn, S. J. Allen, M. Thomas, H. Kroemer, "Midinfrared studies of the contact region at superconductor-semiconductor interfaces", Applied Physics Letters 76, 215-217 (2000). DOI 10.1063/1.125706, https://doi.org/10.1063/1.125706
  15. F. Rohlfing, G. Tkachov, F. Otto, K. Richter, D. Weiss, G. Borghs, C. Strunk, "Doppler shift in Andreev reflection from a moving superconducting condensate in Nb/InAs Josephson junctions", Physical Review B 80, 220507 (2009). DOI 10.1103/PhysRevB.80.220507, https://doi.org/10.1103/PhysRevB.80.220507
  16. Shlomi Bouscher, Roni Winik, Alex Hayat, "Andreev reflection enhancement in semiconductor-superconductor structures", Physical Review B 97, 054512 (2018). DOI 10.1103/PhysRevB.97.054512, https://doi.org/10.1103/PhysRevB.97.054512
  17. Serhii Volkov, M. Gregor, T. Plecenik, Elena Zhitlukhina, Mikhail Belogolovskii, A. Plecenik, "Above-gap differential conductance dips in superconducting point contacts", Applied Nanoscience 12, 761-768 (2022; published online 22 February 2021). DOI 10.1007/s13204-021-01734-6, https://doi.org/10.1007/s13204-021-01734-6
  18. M. Behet, R. Hövel, A. Kohl, A. Mesquida Küsters, B. Opitz, K. Heime, "MOVPE growth of III-V compounds for optoelectronic and electronic applications", Microelectronics Journal 27, 297-334 (1996). DOI 10.1016/0026-2692(95)00060-7, https://doi.org/10.1016/0026-2692(95)00060-7