Phase-driven collapse of the Cooper condensate in a nanosized superconductor

Superconductivity can be understood in terms of a phase transition from an uncorrelated electron gas to a condensate of Cooper pairs in which the relative phases of the constituent electrons are coherent over macroscopic length scales. The degree of correlation is quantified by a complex-valued order parameter, whose amplitude is proportional to the strength of the pairing potential in the condensate. Supercurrent-carrying states are associated with nonzero values of the spatial gradient of the phase. The pairing potential and several physical observables of the Cooper condensate can be manipulated by means of temperature, current bias, dishomogeneities in the chemical composition, or application of a magnetic field. Here we show evidence of complete suppression of the energy gap in the local density of quasiparticle states (DOS) of a superconducting nanowire upon establishing a phase difference equal to $\pi$ over a length scale comparable to the superconducting coherence length. These observations are consistent with a complete collapse of the pairing potential in the center of the wire, in accordance with theoretical modeling based on the quasiclassical theory of superconductivity in diffusive systems. Our spectroscopic data, fully exploring the phase-biased states of the condensate, highlight the profound effect that extreme phase gradients exert on the amplitude of the pairing potential. Moreover, the sharp magnetic response (up to 27 mV/${\mathrm{\Phi }}_0$) observed near the onset of the superconducting gap collapse regime is exploited to realize magnetic flux detectors with noise-equivalent resolution as low as 260 $\mathrm{n}{\mathrm{\Phi }}_0/\sqrt{\text{Hz}}$.

Figure 1

Principle of operation and interferometer design. (a) Schematic representation of the measurement setup for the transport spectroscopy of a phase-biased superconducting wire. The current-voltage characteristics show the modulation of the local density of states in the latter as a function of the magnetic flux $\mathrm{\Phi }$ coupled to a compact superconducting loop in clean contact with the wire. (b) Conceptual picture of the progressive depairing in the middle of the superconducting wire for increasing phase gradient. The position-dependent value of the complex order parameter $\mathrm{\Delta }\left(x\right)exp\left[i\varphi \left(x\right)\right]$ is shown as a twisted-wireframe representation of a revolution surface. For a superconducting wire having single-valued current-to-phase relationship, $\mathrm{\Delta }(L/2)=0$ is expected for $\mathrm{\Phi }={\mathrm{\Phi }}_0/2$. (c) and (d) Pseudocolor scanning electron micrographs of, respectively, the interferometer loop and the superconducting wire in a typical device. Here yellow indicates the 150-nm-thick “interferometer” and 25-nm-thick “nanowire” Al layers; the 15-nm-thick AlOx tunnel probe electrode is shown in purple, realized by oxidizing respectively a Al/${\mathrm{Al}}_{0.98}{\mathrm{Mn}}_{0.02}$ layer to obtain a superconducting/normal-metal electrode.

Figure 2

Magnetoelectric response of a typical normal-metal tunnel probe device. Experimental traces are color coded to six applied magnetic flux values equally spaced in the range $\mathrm{\Phi }=0\to {\mathrm{\Phi }}_0/2$. (a)–(d) Current-vs-voltage characteristic curves, recorded at lattice temperature $T=20$, 300, 500, and 700 mK, respectively. (e) Normalized differential conductance as a function of voltage bias, modeled for lattice temperature $T=20\mathrm{mK}$ (see text). (f) and (g) Differential conductance as a function of voltage bias, recorded at lattice temperature $T=20$ and 650 mK, respectively.

Figure 3

Results for the modelization of the superconducting nanowire according to parameters inferred from experimental data. (a) Circulating supercurrent $I_s$ normalized to the zero-temperature value of the critical current of the wire ($I_c$). (b) Interferometer free energy $F$ normalized to $E_0=I_c{\mathrm{\Phi }}_0/2\pi$. (c) Pairing potential amplitude $\mathrm{\Delta }(x=L/2)$ in the center of the superconducting nanowire normalized to the zero-temperature value of the pairing potential amplitude in the interferometer loop (${\mathrm{\Delta }}_0$). In (a)–(c) different traces are color coded to the temperature values $T=20$, 300, 500, and 700 mK and show the modulation of the respective quantity as a function of the magnetic flux $\mathrm{\Phi }$ applied to the interferometer. Dotted branches in (a) and (b) represent theoretical solutions corresponding to thermodynamically unstable interferometer states. (d)–(g) Normalized differential conductance of the normal-metal tunnel probe device as a function of the applied magnetic flux and voltage bias $V_b$ at lattice temperature $T=20$, 300, 500, and 700 mK, respectively. The halves of each color map plot allow the comparison between theoretical predictions (left) and experimental data (right).

Figure 4

Superconducting tunnel probe spectroscopy of the phase-driven collapse of the superconducting gap in the Al nanowire. (a) and (b) Normalized differential conductance as a function of the applied magnetic flux and voltage bias, recorded at lattice temperature $T=0.9$ and 1.0 K, respectively. The region showing negative differential conductance is indicated in magenta. (c) and (d) Current-vs-voltage characteristic curves recorded for $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.48$, 0.49, and 0.5 and for lattice temperature $T=0.9$ and 1.0 K.

Figure 5

Magnetoelectric response of a typical superconducting tunnel probe device. (a) and (b) Respectively, voltage response (arbitrary offset) and corresponding flux-to-voltage transfer function for $\mathrm{\Phi }\approx {\mathrm{\Phi }}_0/2$, recorded at temperature $T=1.0$ K. The device is here operated under fixed current bias values $I_b=4.30$, 4.35, and 4.40 nA.

Figure 6

Summary of the magnetoelectric response figure of a typical superconducting-probe device recorded at temperature $T=1$ K and current bias $I_b=4.35\mathrm{nA}$. The data sets shown here expand on the corresponding data presented in Fig. 4. (a)–(c) A function of the applied flux $\mathrm{\Phi }$, respectively, the dc voltage, the flux-to-voltage transfer function (obtained by numerical differentiation), and the differential conductance (estimated by finite differences from the voltage traces recorded for $I_b=4.3$, 4.35, and 4.4 nA). The spectral characteristics of the amplified signal (green/blue traces: individual-channel power spectral density ${\mathcal{A}}_{1,2}$, black trace: cross-spectral density ${\mathcal{X}}_{12}$, gray shading: readout noise level estimate) are presented in (d)–(f) for three illustrative flux working points. (d) Zero first-order response ($\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.5$), while (a) and (f) are relative to the device being tuned for highest responsivity and best noise-equivalent flux resolution, respectively.

Figure 7

Cross-spectral density ${\mathcal{X}}_{12}$ (red trace) and readout noise estimate (gray shading) in magnetic flux units. (a) and (b) The device tuned for highest responsivity ($\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.502$) and for best noise-equivalent flux resolution ($\mathrm{\Phi }/{\mathrm{\Phi }}_0=0.5025$), respectively.