Negative nonlocal and local voltages (resistances) in a quasi-one-dimensional superconducting aluminum structure

Abstract: To study a nonlocal electron transport in an aluminum superconducting quasi-one-dimensional structure, we measured negative nonlocal (local) direct current voltages in the structure in a magnetic field near the critical temperature. The structure is a normal-superconducting at $T_{cn}<T<T_{cw}$ ($T_{cn}$ and $T_{cw}$ are the critical temperatures for narrow and wide wires, respectively, making up this structure). Negative voltage arises due to a quasiparticle current flowing through the N-S interface. We plotted the experimental and theoretical temperature and magnetic-field dependences of current, resistance and voltage corresponding to the peak of negative voltage, taking into account either equilibrium or nonequilibrium superconducting fluctuations.

• The paper observes negative direct-current voltages in a quasi-one-dimensional superconducting aluminum structure, reporting both nonlocal and local voltage peaks of -0.69 microvolts and -0.31 microvolts, respectively.
• Key findings include that the quasi-particle injection through a normal-superconducting (N-S) interface produces voltages of sign opposite to the applied current, with negative resistances of -1.16 Ohms and -0.56 Ohms measured.
• The observed voltage and resistance phenomena are driven by a quasiparticle charge imbalance, and examined temperature and magnetic field dependencies.

Overview

Kuznetsov and Trofimov report the observation of negative direct-current voltages—both nonlocal and local—in a quasi-one-dimensional aluminum structure composed of narrow and wide wires with different critical temperatures. The measurements were performed at temperatures very close to $T_c$ and in perpendicular magnetic fields, using an X-Y plotter-based analog setup designed to suppress electromagnetic interference (input-referred noise of 0.03 μV peak-to-peak). The central claim is that a quasiparticle current injected through a normal-superconducting (N-S) interface produces a voltage of sign opposite to the applied current, corresponding to $\bar{\mu}_q < \bar{\mu}_p$, i.e., a negative difference between the electrochemical potentials of quasiparticles and Cooper pairs.

Structure and experimental parameters

The structure was fabricated by thermal deposition of 19 nm-thick aluminum on silicon via electron-beam lithography lift-off. It consists of a wide wire ($w_w = 0.50$ μm) flanked by two narrow wires ($w_n = 0.27$ μm), separated by $L_{21} = 6.69$ μm, chosen so that $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$, where $\lambda_Q$ is the quasiparticle charge-imbalance relaxation length. Key parameters: $R_0 = 37.85\ \Omega$, $R_{sq} = 2.83\ \Omega$, $T_c = 1.486$ K, $\bar{\mu}_q < \bar{\mu}_p$0 nm, $\bar{\mu}_q < \bar{\mu}_p$1 m²/s, $\bar{\mu}_q < \bar{\mu}_p$2 μm. The structure satisfies $\bar{\mu}_q < \bar{\mu}_p$3, confirming quasi-one-dimensionality.

The resistive N-S transition $\bar{\mu}_q < \bar{\mu}_p$4 was fitted with Aslamazov-Larkin (AL) corrections alone and with AL plus anomalous Maki-Thompson (MT) corrections; both fits are adequate, but the fitted depairing factor $\bar{\mu}_q < \bar{\mu}_p$5 exceeds the calculated value $\bar{\mu}_q < \bar{\mu}_p$6 by a factor of 6.9, which the authors attribute to unaccounted phase-breaking mechanisms such as external noise—a limitation acknowledged but not resolved.

Observation of negative voltages

At $\bar{\mu}_q < \bar{\mu}_p$7 K in zero field, two measurement circuits yield negative voltages: the nonlocal circuit ($\bar{\mu}_q < \bar{\mu}_p$8: probes 3–2; $\bar{\mu}_q < \bar{\mu}_p$9: probes 1–4) shows a peak of −0.69 μV at $w_w = 0.50$0 μA, and the local circuit ($w_w = 0.50$1: 3–1; $w_w = 0.50$2: 2–4) peaks at −0.31 μV at $w_w = 0.50$3 μA. The corresponding resistances are $w_w = 0.50$4 and $w_w = 0.50$5. Crucially, negative voltage appears only when the current-carrying part contains both a superconducting and a normal wire, and the voltage pair consists of one normal and one superconducting wire. Circuits where this condition fails show only conventional positive resistive response above $w_w = 0.50$6. The curves are odd functions of current without hysteresis, and the peak currents coincide with independently measured critical currents, supporting their identification as critical currents of the N-S structure.

Origin: heterogeneous N-S structure

The authors establish that the narrow and wide wires have different critical temperatures, $w_w = 0.50$7–1.460 K and $w_w = 0.50$8 K, verified by reference wires ($w_w = 0.50$9 K for a narrow wire; 1.447 vs 1.490 K for separate narrow/wide wires). This yields a notable and somewhat counterintuitive result: the critical temperature decreases by about 40 mK when the wire width shrinks from 0.5 to 0.27 μm, contradicting the expected scaling $w_n = 0.27$0 inferred from thin-film data. The proposed explanation invokes depairing centers (magnetic impurities, vacancies) concentrated at dirty lateral boundaries, whose relative influence grows as the wire narrows. This explanation remains speculative; no independent characterization of boundary disorder is presented.

Within $w_n = 0.27$1, the structure is effectively a heterogeneous N-S system. Quasiparticles injected from the normal narrow wire into the superconducting wide wire create local or nonlocal charge imbalance. Following the SBT framework, the measured voltage is identified with

$w_n = 0.27$2

at $w_n = 0.27$3 μm from the N-S interface center, since $w_n = 0.27$4 there. With $w_n = 0.27$5 μm, the SBT prediction gives $w_n = 0.27$6, which matches the measured values only in order of magnitude and only over part of the temperature range.

Temperature dependences

The temperature dependences of the negative resistances are nonmonotonic and cannot be captured by the nonequilibrium SBT model alone across the full range 1.453–1.491 K. Fits incorporating equilibrium superconducting fluctuations above $w_n = 0.27$7 (AL + MT corrections) describe the data qualitatively below ~1.484 K, while near $w_n = 0.27$8 the nonlocal resistance follows a linear law vanishing at $w_n = 0.27$9, with $L_{21} = 6.69$0. Two striking empirical regularities emerge:

• Near $L_{21} = 6.69$1, the negative nonlocal resistance is directly proportional to the nonlocal critical current, $L_{21} = 6.69$2 with $L_{21} = 6.69$3A.
• In magnetic field, $L_{21} = 6.69$4 with $L_{21} = 6.69$5 and $L_{21} = 6.69$6A at the two temperatures studied.

The critical currents themselves follow linear rather than GL $L_{21} = 6.69$7 laws in the range 1.455–1.491 K; GL fits give unphysical fitting temperatures above $L_{21} = 6.69$8 and critical currents well below $L_{21} = 6.69$9 μA. The linear behavior is interpreted via formation of a Josephson junction at the N-S connection, with fitted junction resistances (25.6 and 38.0 Ω) mapping onto plausible wire lengths. Below 1.4 K, where negative voltages vanish, the critical current reverts to standard GL behavior with $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$0 μA close to the depairing value—an internal consistency check supporting the interpretation.

Magnetic-field dependence

At fixed temperature, all field-dependent quantities—the nonlocal critical current, negative resistance, and peak voltage—follow GL-like power laws in $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$1: the resistance scales as $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$2 and the voltage as the cube of that factor, with fitted coherence lengths $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$3 μm and $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$4 μm close to calculated values. The SBT model reproduces the field dependence of the resistance in order of magnitude for $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$5 G but fails near $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$6. The proportionality between negative resistance and the GL depairing critical current in field is reported as unexpected and is not derived from first principles.

Limitations and open questions

Several limitations are conceded explicitly. First, no unified model exists: the equilibrium fluctuation theory and the nonequilibrium SBT model each capture parts of the phenomenology, and the authors state that a combined treatment—including the influence of the potential-pair wires—is required. Second, the discrepancy between fitted and calculated depairing factors suggests residual phase-breaking not accounted for. Third, the mechanism proposed for $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$7 (boundary depairing centers) is conjectural, and the claimed validity of $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$8 for $\xi(T) \ll L_{21} \approx \lambda_Q(T,B)$9 is asserted without measurement. Fourth, the proportionality constants linking negative resistance to critical current lack theoretical derivation. Finally, the distinction between the nonlocal and local curves is attributed only to the small difference $\lambda_Q$0 between the two narrow wires, supported by asymmetric measurements from opposite sides of the structure but not quantitatively modeled.

Conclusion

This work demonstrates that negative nonlocal and local dc voltages arise in a heterogeneous quasi-one-dimensional Al structure within the window $\lambda_Q$1, driven by quasiparticle injection through N-S interfaces and measurable as a negative difference $\lambda_Q$2. The effect requires a mixed normal/superconducting probe configuration, vanishes at $\lambda_Q$3, and exhibits empirical proportionality between negative resistance and critical current in both temperature and field. The principal open problem left by the paper is the construction of a theoretical framework that treats equilibrium superconducting fluctuations above $\lambda_Q$4 and nonequilibrium charge imbalance below $\lambda_Q$5 simultaneously, so that the full temperature and field dependences—and the observed resistance–critical-current proportionality—can be derived rather than fitted.