A quantitative theory of the anomalous quasiparticle transport in Nb-contacted InAs-AlSb quantum wells: the giant precursor of induced superconductivity and sub-flux-quantum-periodic resistance oscillations

InAs-AlSb quantum wells contacted by niobium electrodes form exceptionally transparent superconductor-semiconductor weak links whose transport is dominated by multiple Andreev reflections (MARs). The 1994 source paper (Kroemer, Nguyen, Hu, Yuh, Thomas, and Wong, Physica B 203, 298) reported two interrelated anomalies in these devices that it could not explain, posing a worked-out theory as an open task. First, above the temperature at which superconductivity is induced in the quantum well, the zero-bias conductance enhancement empirically follows a fluctuation-like law $\Delta G = G_0 T_c/(T-T_c)$ whose prefactor exceeds the standard Aslamazov-Larkin thin-film prediction by about four orders of magnitude. Second, when induced superconductivity in periodic Nb grating arrays is destroyed by weak perpendicular magnetic fields, the resistance oscillates with periods of 4.7 microtesla (sample A) and 10 microtesla (sample B) instead of the canonical 21.8 microtesla corresponding to one flux quantum per grating cell; the authors hypothesized a self-organized 'flux cell superlattice'. The subsequent literature shows: the same group's follow-ups (Phys. Rev. B 54, R2311 (1996); Phys. Rev. B 58, 11676 (1998)) could not reproduce the anomalous oscillation periods, explained the reproducible canonical-period oscillations by staggered vortex superlattices, and left the giant precursor explicitly outside established theories; the one dedicated MAR-theory effort (Jacobs, Kümmel, and Plehn, 1999) reproduces only some current-voltage features. Both pillars of the source question remain without a quantitative theory, and the sub-flux-quantum periods remain unexplained and unreproduced.

Background

InAs-AlSb quantum wells confine electrons in a 15-nm-wide indium arsenide layer between antimony-based barriers, forming a two-dimensional electron gas (2DEG: a sheet of conduction electrons free to move in the plane but quantum-confined perpendicular to it) with sheet density $n_s \approx 8\times10^{12}\,\mathrm{cm^{-2}}$ and mobility $\approx 8\times10^{4}\,\mathrm{cm^2/Vs}$ at 4.2 K. The surface Fermi level of InAs is pinned inside the conduction band, so metal contacts to InAs form no Schottky barrier (the electrically insulating depletion layer that normally forms at a metal-semiconductor interface). Niobium (Nb) electrodes deposited on the quantum well therefore make exceptionally transparent superconductor-semiconductor contacts, and superconducting correlations spread from the Nb into the adjacent normal region: the superconducting proximity effect, here called induced superconductivity.

The elementary charge-transfer process at such an interface is Andreev reflection (AR): an electron in the normal region incident on a superconductor with energy below the superconducting gap cannot enter as a quasiparticle and is reflected as a hole in the time-reversed state, transferring a Cooper pair (two electrons bound into the superconducting condensate, charge $2e$) into the superconductor. In a superconductor-normal-superconductor (SNS) weak link (a normal region connecting two superconducting electrodes), an electron can be Andreev-reflected at one electrode, travel back toward the other as a hole, be Andreev-reflected again into an electron, and so on. Such multiple Andreev reflections (MARs) convert quasiparticle transport into Cooper-pair transfer: in phase-coherent form they carry the supercurrent at zero voltage, and at finite voltage they produce the subharmonic gap structure, conductance peaks at bias voltages $V = 2\Delta/n$ for integers $n$, where $\Delta$ is the superconducting energy gap.

Kroemer and co-workers studied Nb electrodes on InAs-AlSb quantum wells (Nguyen, Kroemer, and Hu, 1992; Kroemer et al., 1994). A specific feature of the 15-nm well is backplane enhancement of AR: electrons that fail to undergo AR at the Nb interface are specularly reflected by the lower AlSb barrier and return to the interface for further attempts, so the effective AR probability is large even though the single-event probability is small ($\alpha \approx 3.5\%$). The average distance an electron travels along the interface before an AR event, the Andreev transfer length, was measured to be $L_A = 1.4$-$1.6\,\mu\mathrm{m}$, exceeding the 0.6-0.75 micrometer width of the Nb lines in the grating structures, so quasiparticles can couple neighboring grating cells. A control sample in which the quantum well is replaced by a 0.5-micrometer homogeneously doped InAs layer (no specular backplane) shows no central zero-bias conductance spike, supporting the MAR interpretation of the conductance enhancement.

The devices exhibiting the anomalies studied here are periodic gratings of Nb lines on the 2DEG: roughly 300 weak links in series, grating period 1 micrometer, array width 95 micrometers. In such a Josephson array (a periodic array of weak links whose collective resistance and supercurrent are governed by interference of the superconducting order parameter across the array), magnetic-interference arguments predict resistance oscillations with one superconducting flux quantum, $\Phi_0 = h/2e \approx 2.07\times10^{-15}\,\mathrm{Wb}$, per grating cell. One $\Phi_0$ per cell of area $1\,\mu\mathrm{m} \times 95\,\mu\mathrm{m}$ corresponds to a field period $\Delta B \approx 21.8\,\mu\mathrm{T}$.

Against this background, the source paper reported two anomalies. (1) Giant precursor of induced superconductivity: above the temperature $T_c \approx 3.9\,\mathrm{K}$ at which superconductivity is induced in the 2DEG, the zero-bias conductance enhancement follows empirically

$$\Delta G = G_0\,\frac{T_c}{T - T_c}, \quad G_0 = 4.5\times10^{-2}\,\Omega^{-1}$$

which has the functional form of the Aslamazov-Larkin (AL) theory, the standard theory of fluctuation paraconductivity (the excess conductance of a superconducting film above its transition caused by thermally fluctuating Cooper pairs; Larkin and Varlamov, 2005). But the prefactor exceeds the AL thin-film prediction $G_0 = (e^2/16\hbar)(b/l) \approx 5\times10^{-6}\,\Omega^{-1}$ for this aspect ratio by roughly four orders of magnitude. The same conductance peak in single-gap devices is quenched by perpendicular magnetic fields in two stages: a sharp drop within $\sim 1\,\mu\mathrm{T}$ (a flux of order one $\Phi_0$ through the gap), attributed to the destruction of phase-coherent MAR trains, followed by a much slower decay attributed to incoherent MARs. The precursor evolves continuously into the supercurrent below $T_c$, where the devices show a Josephson-type current-voltage characteristic with $I_c R_n \approx 1\,\mathrm{mV}$. (2) Anomalous flux-periodic resistance oscillations: when induced superconductivity in the gratings is destroyed by a perpendicular field of order tens of microtesla, the reestablished resistance contains a periodic component with periods $\Delta B \approx 4.7\,\mu\mathrm{T}$ (sample A, about $0.22\,\Phi_0$ per cell) and $10\,\mu\mathrm{T}$ (sample B, about $0.46\,\Phi_0$) instead of the canonical $21.8\,\mu\mathrm{T}$, with temperature-independent oscillation frequency and phase (Fourier quality factor $Q \approx 8.2$) and quenching by DC bias currents of order $10\,\mu\mathrm{A}$. The authors hypothesized a flux cell superlattice: a self-organized periodic arrangement of flux cells, each larger than the grating unit cell and carrying an integer number of flux quanta, with enhanced pinning (immobilization of vortices, the quantized magnetic-flux filaments threading a superconductor or weak link, by defects) when commensurate with the grating. No direct observation (for example magnetic imaging) of such a structure was available.

What standard physics predicts for such arrays became clearer in the group's later samples: canonical $\Phi_0$-per-cell oscillations above a few hundred microtesla, and at lower fields in wide arrays a frequency doubling attributable to a staggered vortex superlattice, the arrangement of Owen-Scalapino modes in long Josephson junctions (junctions longer than the Josephson penetration depth, the length scale over which phase differences vary within the junction, which support Josephson vortices, flux quanta trapped in the junction interior rather than in the electrodes; Owen and Scalapino, 1967). Whether standard mechanisms of this kind can account for the 1994 phenomenology, or whether the anomalous periods indicate a distinct or sample-specific effect, remains open.

Problem Statement

The problem, as posed by Kroemer, Nguyen, Hu, Yuh, Thomas, and Wong (Physica B 203, 298 (1994)) and left unresolved by the subsequent literature, is to work out a quantitative theory that accounts for the anomalous quasiparticle-transport phenomenology of Nb-contacted InAs-AlSb quantum-well devices. Its two linked pillars are:

(a) The giant precursor of induced superconductivity. A theory must account for the zero-bias conductance enhancement above the induced transition that empirically follows $\Delta G = G_0\,T_c/(T-T_c)$ with $G_0 = 4.5\times10^{-2}\,\Omega^{-1}$, about $10^4$ times the Aslamazov-Larkin thin-film value, including its continuity with the supercurrent below $T_c$ (Josephson-type characteristic with $I_c R_n \approx 1\,\mathrm{mV}$), its two-component quenching by perpendicular fields of order $1\,\mu\mathrm{T}$ (a sharp drop attributed to destruction of phase-coherent MAR trains, then a slow decay), and its absence in the homogeneously doped control sample without the quantum-well backplane. It must also establish the correct fluctuation baseline for a series array of MAR-coupled weak links, since the AL formula used as the reference in the source applies to a continuous film.

(b) The anomalous flux-periodic resistance oscillations. An account is needed of the periods $4.7\,\mu\mathrm{T}$ (sample A) and $10\,\mu\mathrm{T}$ (sample B) versus the canonical $21.8\,\mu\mathrm{T}$ per grating cell (about $0.22\,\Phi_0$ and $0.46\,\Phi_0$ per cell), with temperature-independent frequency and phase, quality factor $Q \approx 8.2$, and quenching by DC bias currents of order $10\,\mu\mathrm{A}$, including an assessment of the flux-cell-superlattice hypothesis (the flux arrangement within each cell, commensurability with the grating, and the role of the Andreev transfer length exceeding the Nb line width).

References

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