Fluctuation-induced dc resistance of ballistic semiconductor-coupled superconducting weak links: interrupted Andreev-reflection statistics and the width-proportional activation energy

In superconducting weak links whose coupling medium is the ballistic two-dimensional electron gas of an InAs quantum well between Nb electrodes about 0.5 µm apart - a separation larger than the superconducting coherence length but far smaller than the elastic mean free path - the zero-bias resistance below the Nb transition is thermally activated, with an apparent activation energy proportional to the device width $b$ ($E/b$ of order 0.04-0.16 meV/µm) and a steepness parameter $\gamma\gtrsim 100$ in $I_s=\gamma e k_B T/\hbar$. This makes the devices roughly two orders of magnitude more dissipative than the equilibrium Ambegaokar-Halperin fluctuation theory predicts for the same threshold current, and the width-proportional activation energy falls outside established theories. The 1998 paper that established these facts proposed, but explicitly declined to develop, an explanation: a fluctuation-induced dc resistance generated by the statistics of multiple-Andreev-reflection trains interrupted by interface normal reflections and inelastic scattering, i.e. the 'giant shot noise' mechanism. The open problem is to construct that quantitative microscopic theory - reproducing the width-proportional activation energy, the $\gamma$ anomaly, and the dependence on electron sheet concentration and interface quality - and to validate it experimentally, including a direct separation of the equilibrium Josephson and nonequilibrium Andreev-reflection contributions to the corner current on single weak links. Later literature verified giant multiple-Andreev-reflection noise in related junctions and established the nonequilibrium nature of the currents, but no quantitative theory of the fluctuation-induced resistance of these long ballistic links exists.

Background

When two superconducting electrodes are joined by a short non-superconducting conduction channel, the result is a superconducting weak link; with a normal-metal channel it is called an SNS junction, and it behaves in many respects like a Josephson junction, a device whose dissipationless supercurrent is set by the quantum-mechanical phase difference between the two superconducting condensates. In the devices studied here the channel is the two-dimensional electron gas (2DEG) of a modulation-doped InAs quantum well with AlSb barriers, and the electrodes are niobium (Nb) stripes separated by $L\approx 0.5\,\mu\mathrm{m}$ - a distance larger than the superconducting coherence length $\xi_n$ in the 2DEG (the scale over which the superconducting order would recover; at most about $0.29\,\mu\mathrm{m}$ at 4.2 K) but much smaller than the elastic mean free path $l_{\mathrm{el}}$ (several micrometers), so quasiparticle motion between the electrodes is ballistic rather than diffusive. InAs is used because its Fermi level at metal interfaces lies inside the conduction band, so no Schottky barrier (a blocking depletion layer) forms at the Nb-InAs contact.

The elementary microscopic process in such junctions is the Andreev reflection (AR): an electron in the semiconductor incident on a superconductor with energy below the superconducting gap is coherently retro-reflected as a hole, while a Cooper pair enters the condensate. Quasiparticles can then bounce repeatedly between the two electrodes, alternating between electron and hole character and transferring one Cooper pair to the opposite electrode at each AR; such a multiple Andreev reflection (MAR) sequence imprints the subharmonic gap structure - oscillations of the differential conductance at voltages $V=2\Delta/ne$, with $\Delta$ the gap and $n$ an integer - on the current-voltage characteristic (CVC). An unbroken MAR sequence is the microscopic origin of the dissipationless supercurrent of ballistic weak links; a train interrupted by a normal (non-Andreev) reflection at either superconductor-semiconductor interface, or by an inelastic scattering event, transfers a random net charge and reverses the direction of pair transfer, so the random interruption of trains is a mechanism for large current fluctuations. For short superconducting quantum point contacts, Averin and Imam showed that such fluctuations constitute 'giant shot noise' of the supercurrent itself, with an effective transferred charge that grows as $1/V$ at low bias.

A Josephson junction biased below its critical current ordinarily shows zero dc voltage. The standard classical description is the resistively shunted junction (RSJ) model, in which an ideal supercurrent branch (the equilibrium Josephson current) is connected in parallel with a resistive quasiparticle branch. Ambegaokar and Halperin (AH) showed that the thermal Johnson noise of the resistive branch makes the junction phase diffuse over its energy barrier already below the critical current, so a current-biased junction exhibits a small, thermally activated dc resistance near zero bias - a fluctuation-induced resistance whose activation energy is set by the Josephson energy and hence by the critical current, which scales with junction width.

Against this background, M. Thomas, H.-R. Blank, K. C. Wong, H. Kroemer, and E. Hu [Phys. Rev. B 58, 11676 (1998)] reported measurements on series arrays of about 300 nominally identical Nb/InAs-quantum-well weak links (arrays were used to raise the measurable voltage; each cell had a width $b=10$-$200\,\mu\mathrm{m}$ perpendicular to the current flow). Below the Nb transition temperature ($T_{\mathrm{Nb}}\approx 8.5$ K) the zero-bias differential resistance $R_0(T)$ falls over several kelvin - orders of magnitude more slowly than in tunnel junctions - and follows an Arrhenius law with an apparent activation energy $E_{\mathrm{app}}$ that is proportional to the device width: a $200\,\mu\mathrm{m}$-wide device does not behave like two $100\,\mu\mathrm{m}$ devices in parallel (which would merely halve the resistance) but has twice the activation energy, with $E/b\approx 0.04$-$0.16$ meV/$\mu$m across samples. The differential resistance below the corner current $I_c$ (the current at the knee of the CVC, itself proportional to width) collapses onto a width-independent function,

$R(I)/R_a = f(I_c - I), \qquad R(I)/R_a \approx \exp(-\delta I/I_s), \qquad I_s = \gamma\, e k_B T/\hbar,$

with $\delta I = I_c - I$, $R_a$ the high-bias asymptotic resistance, and a dimensionless steepness parameter $\gamma\gtrsim 100$. The activation energy is then $E_{\mathrm{app}}\approx \hbar I_c/(\gamma e)$, i.e. a factor $\gamma\gtrsim 100$ smaller than the scale $\hbar I_c/e$ that an equilibrium AH-type theory with the same threshold current would produce. Concretely, force-fitting the AH theory to the measured CVC of a $65\,\mu\mathrm{m}$ device yields a critical current within about 10% of 55 µA and a predicted activation energy of about 226 meV, a factor of about 87 above the measured 2.6 meV: the devices are far more dissipative than equilibrium Johnson-noise fluctuations allow.

The authors concluded that the equilibrium AH/RSJ framework cannot describe ballistic weak links in this intermediate-temperature regime, because there is no clean separation between an ideal supercurrent branch and a resistive noisy branch - the phase coupling itself is carried by MAR, and its random interruptions generate nonequilibrium noise. They proposed a fluctuation-induced dc resistance based on the statistics of interrupted MAR trains (the giant-shot-noise mechanism) as the promising route, while explicitly declining to develop it. They also identified related unresolved issues: the apparent activation energy appears to increase with the 2DEG sheet electron concentration $N_s$, but the trend cannot be separated from uncontrolled processing-induced variations of the Nb-InAs interface quality (which can change the interface scattering from specular to diffusive); and the measurable corner current may contain, besides the equilibrium Josephson current, a large nonequilibrium AR-driven component - as suggested by Shapiro steps (voltage plateaus in the CVC under microwave irradiation, at $V_n = n h f/2e$ for drive frequency $f$) appearing at half the canonical voltage and persisting to temperatures where the dc supercurrent has vanished.

Problem Statement

Provide a quantitative, microscopic account of the fluctuation-induced dc resistance of ballistic semiconductor-coupled superconducting weak links in the regime $\xi_n < L < l_{\mathrm{el}}$ (as realized by the Nb/InAs-quantum-well devices with $L\approx 0.5\,\mu\mathrm{m}$ and widths $b=10$-$200\,\mu\mathrm{m}$), and validate it experimentally. The problem, as posed by the source work, comprises four jointly supporting parts: (1) construct the theory - starting from the multiple-Andreev-reflection dynamics of ballistic quasiparticles between two superconducting electrodes, including the interface normal-reflection probabilities and inelastic scattering, compute the statistics of interrupted Andreev-reflection trains, the resulting current-noise spectrum, and how these nonequilibrium fluctuations are rectified into a measurable dc resistance below the corner current; (2) reproduce the empirical laws - the thermally activated zero-bias resistance $R_0(T)\propto\exp(-E_{\mathrm{app}}/k_BT)$ with $E_{\mathrm{app}}\propto b$ (with $E/b\approx 0.04$-$0.16$ meV/µm over the explored widths), the width-collapsed form $R(I)/R_a = f(I_c-I)$ with steepness current $I_s=\gamma e k_B T/\hbar$ and $\gamma\gtrsim 100$, and the excess of the observed resistance over the equilibrium Ambegaokar-Halperin prediction by roughly two orders of magnitude for the same threshold current; (3) predict the parameter dependences - how $E_{\mathrm{app}}$ (and $E/b$) vary with the 2DEG sheet electron concentration $N_s$ and with the processing-dependent Nb-InAs interface transparency/specularity, and which measurements would disentangle these from uncontrolled interface variations; (4) experimentally separate the current composition - on single (non-array) weak links with lower noise floors, directly determine the relative magnitudes of the equilibrium Josephson contribution and the nonequilibrium AR-driven contribution to the corner current, and measure the noise spectrum to identify the fluctuation source (equilibrium thermal noise versus multiple-Andreev-reflection shot noise). The theory is to reproduce the enumerated empirical laws, including the width scaling; part (2) is the empirical test of the theory of part (1). Part (4) requires no new measurements: it consists of the single-link measurement protocol and a quantitative prediction of its outcome - the relative magnitudes of the equilibrium Josephson and nonequilibrium Andreev-reflection contributions to the corner current, and the noise spectrum that identifies the fluctuation source.

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