Letter | Published:

Collective energy gap of preformed Cooper pairs in disordered superconductors

Abstract

In most superconductors, the transition to the superconducting state is driven by the binding of electrons into Cooper pairs1. The condensation of these pairs into a single, phase-coherent, quantum state takes place at the same time as their formation at the transition temperature, Tc. A different scenario occurs in some disordered, amorphous, superconductors: instead of a pairing-driven transition, incoherent Cooper pairs first preform above Tc, causing the opening of a pseudogap, and then, at Tc, condense into the phase-coherent superconducting state2,3,4,5,6,7,8,9,10,11. Such a two-step scenario implies the existence of a new energy scale, Δc, driving the collective superconducting transition of the preformed pairs2,3,4,5,6. Here we unveil this energy scale by means of Andreev spectroscopy5,12 in superconducting thin films of amorphous indium oxide. We observe two Andreev conductance peaks at ±Δc that develop only below Tc and for highly disordered films on the verge of the transition to insulator. Our findings demonstrate that amorphous superconducting films provide prototypical disordered quantum systems to explore the collective superfluid transition of preformed Cooper pairs.

Main

While a small amount of non-magnetic disorder weakly affects superconductivity13,14, the situation changes radically when disorder reaches a level such that localization of electronic wavefunctions develops. In some amorphous materials, such an extreme situation in which the normal-state electron transport is no longer metallic due to the abundant scattering yields a fragile superconducting state at low temperature. On increasing disorder further or applying a magnetic field, this superconducting state undergoes a sharp quantum phase transition to insulator15,16.

Recently, a body of tunnelling experiments performed on different disordered thin films7,8,9,10,11,17,18,19 have shown that the spectral properties of superconductivity bordering the transition to insulator deviate significantly from the Bardeen–Cooper–Schrieffer (BCS) theory1. The new picture drawn by these measurements is that electrons preform into Cooper pairs10 well above the transition temperature, Tc, leading to a deep suppression of the single-particle density of states at the Fermi level, which is known as the pseudogap anomaly8,9,10. Only at a temperature T = Tc at which resistivity vanishes and global phase coherence between these preformed pairs sets in, the pseudogap evolves into a hard gap, Eg. The systematic observation of an anomalously large and spatially fluctuating spectral gap Eg (see refs 7,8,9,10,11), with a ratio Eg/kBTc reaching up to approximately 6 in the most disordered samples10, has pointed out a dissociation between Tc and Eg, in stark contrast to BCS superconductors for which Eg/kBTc = 1.76 (kB is the Boltzmann constant).

On these grounds, theory predicts that a second energy scale, Δc, may be at play to drive the collective, BCS-type, superconducting transition of the preformed pairs2,3,4,5,6. This collective gap is expected to scale with Tc and thus lies within the single-particle gap Eg, making it impossible to detect in a tunnelling measurement.

As earlier proposed in the context of high-Tc superconductors12,20, the means to unveil Δc consists of coherent transfer of pairs of electrons from a normal metal (N) electrode to the superconductor (S). The mechanism that allows charge transfer through the N/S interface correlates, in the metal, an impinging electron with another one, having energy below the Fermi level, to form a Cooper pair in the superconductor. This so-called Andreev process results from the particle–hole mixing occurring in the superconductor within the energy interval of the BCS superconducting gap21. On the other hand, in a superconductor with preformed Cooper pairs, theory predicts that the energy window for particle–hole mixing no longer coincides with the single-particle gap Eg but instead with Δc (ref. 5). Consequently, whereas single-particle tunnelling probes only Eg, two-particle spectroscopy, termed Andreev spectroscopy, enables direct measurement of the collective energy gap Δc.

Here we report on Andreev spectroscopy performed on superconducting thin films of amorphous indium oxide on the verge of the insulating state22. With the transfer of electron pairs being a second-order process as compared to single-electron tunnelling, Andreev spectroscopy requires high contact transparency. We therefore used the metallic tip of a scanning tunnelling microscope (STM) to control the transparency, that is, the conductance of the point-contact formed between the tip and the film, and performed a systematic study from single-particle tunnelling to Andreev spectroscopy23.

The chief result of this work is presented in Fig. 1 where we display a set of differential conductance curves G(V) = dI/dV versus bias-voltage V across a point contact on sample InO-2. For each curve, the point-contact conductance is controlled by adjusting the current set-point of the STM feedback loop for a given voltage bias (V = −3 mV). From the bottom to the top curve, the point-contact conductance varies from e2/h to more than 4e2/h (e is the electron charge; h is the Plank constant).

We begin by noting that for the low point-contact conductance (bottom curve in Fig. 1), G(V) resembles that of single-particle tunnelling with a gap Eg at the Fermi level bordered by single-particle peaks (see Supplementary Fig. 2). Increasing further the conductance profoundly modifies the spectrum. While the single-particle peaks remain, the gap fills in with a significant level of conductance indicating the opening of new conduction channels for two particles as the transparency rises. The Andreev spectroscopy regime is hence clearly established.

For conventional, low-disorder superconductors, the Andreev process at the interface leads to a monotonous increase of conductance inside the gap ultimately reaching, for perfect transparency, twice the contact conductance24. Our results on a highly disordered film displayed in Fig. 1 are rather different. Two peaks inside the single-particle gap emerge out of the Andreev conductance background as the contact conductance rises above 2e2/h. Their energies are symmetric with respect to zero bias, at ±150 μV, and independent of the contact transparency. These new peaks unveiled by Andreev spectroscopy are the main focus of this work and we shall demonstrate that they provide compelling evidence for the collective gap Δc for the preformed Cooper pairs as predicted by the theory in ref. 5.

To trace back the superconductivity-related origin of these peaks, we studied the evolution of the Andreev spectra as a function of T measured at different point-contact locations of the samples. One representative T evolution of G(V) measured on sample InO-2 is shown in Fig. 2a. The point contact transparency is set large enough such that both Andreev and single-particle peaks are apparent. As the temperature increases, the two Andreev peaks move towards zero bias voltage at T 0.6Tc and merge further into a conductance maximum at zero bias. This maximum, still just visible at T = Tc, disappears beyond T 1.05Tc. The T evolution of the Andreev peaks is seen even more clearly in Fig. 2b by plotting −d2G/dV2 versus T and V (see also Supplementary Fig. 8).

The termination of the Andreev peaks at TTc that we observe in all of our samples clearly indicates that they relate to coherent superconductivity5. This is in stark contrast with the T evolution of the single-particle tunnelling gap Eg, which remains unchanged at Tc, and only vanishes further at several times Tc, resulting in the appearance of the pseudogap in the single-particle density of states10.

From a theoretical standpoint, the body of work on disordered superconductivity has reached a consensus in explaining the pseudogap by the preformation of Cooper pairs above Tc (refs 4,5,6,25). The mechanism behind this relies on the delicate interplay between quantum localization of the single-particle states and the attractive pairing. As a result, the anomalously large single-particle gap, Eg, is predicted to embody two contributions. The first is the pairing energy gap Δp for the preformation of Cooper pairs—the energy gain to pair electrons or equivalently the energy cost to add an odd number of electrons in localized orbitals—at the origin of the pseudogap. The second is the collective energy gap Δc related to the BCS-type condensation of the preformed pairs into the superconducting state5. As Δp is not involved when transferring two electrons simultaneously, only Δc can give rise to the Andreev conductance signal that we observe in our data.

Furthermore, we investigate how the values of Eg and Δc spread over three highly disordered samples (samples InO-1, InO-2 and InO-3). All of our measurements of Eg and Δc performed on these samples are summarized in Fig. 3. The value of the local gap Eg obtained by fitting the density of states of spectra in the tunnelling regime (see Supplementary Figs. 2 and 3) is strongly dependent on the position of the tip with an energy varying by a factor of more than 3, that is, spreading between 200 μeV and 650 μeV, as previously reported10. In contrast, Δc scatters around an average value of 145 μeV with a variability of ±30 μeV that stems mainly from the experimental accuracy. Note that this observation rules out multiple Andreev reflections at sub-multiple energies 2Eg/n, n being an integer, as a possible origin for the observed sub-gap structure in our spectra26,27. Importantly, this average value of Δc that translates to 1.7 ± 0.3 K is close to Tc in all 3 samples (1.45, 1.5 and 1.9 K; see Supplementary Table 1).

The two energy scales discussed in this work span two clearly distinct ranges that are illustrated in Fig. 4 where we report the respective energy ranges for Δc and Eg on a typical resistance curve around the transition to the superconducting state. The condensation energy Δc is of the order of the critical temperature Tc, in stark contrast to the pseudogap Eg, which extends up to approximately 6Tc. This coincidence between Δc and Tc together with the suppression of Δc in the close vicinity of Tc provides strong evidence that this new energy scale relates to the set in of macroscopic, superconducting phase coherence.

Finally, to demonstrate that the Andreev peaks stem from the high level of disorder in our films and the proximity to the insulator, we conducted identical measurements on a low-disorder amorphous indium oxide film (sample InO-4) that behaves in all points as a conventional dirty superconductor28, that is, without a pseudogap above Tc (see Supplementary Fig. 10b). The corresponding Andreev spectroscopy shows a standard, monotonous increase of the conductance within the single-particle gap, without any intra-gap peak (see Supplementary Fig. 11). This evolution, which has been observed for any position of the STM tip, can be accurately described by Blonder–Tinkham–Klapwijk theory for N/S interfaces24.

To conclude, the scenario drawn by our observations contrasts with standard BCS superconductors. The second energy scale we uncover together with the pseudogap complies with a two-step transition to superconductor, which develops, first, far above Tc with the preformation of Cooper pairs and follows at lower T by the collective condensation of the preformed pairs into a macroscopically phase-coherent superconducting state. The nature of such a superconducting state that develops in disordered superconductors bordering insulators defies conventional theories and shall inspire new theoretical paradigms for superconductivity.

Methods

Samples

Our samples are disordered thin films of amorphous indium oxide. The films are prepared by electron-beam evaporation of high-purity (99.99%) In2O3 onto SiO2 in an O2 partial pressure. Samples were patterned into Hall bridges via a shadow mask to perform in situ four-terminal transport measurements in the STM set-up and thus direct comparison between macroscopic and microscopic superconducting properties.

Measurements

Tunnelling and Andreev spectroscopy measurements were performed with a home-built STM cooled down to 0.05 K in an inverted dilution refrigerator. By varying the STM set-point current from 1 nA to 1 μA with a bias voltage of a few millivolts, we tuned the junction resistance in the range 1–1,000 kΩ and thus explored continuously the whole transition from the tunnelling to the contact regime. The differential conductance $$G(V)$$ = $${{{\rm d}I}/{{\rm d}V}}$$ of the junction was measured with a lock-in amplifier technique with a voltage modulation of 10–30 μV.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank M. Feigel’man, L. Ioffe and Y. Nazarov for fruitful discussions. This research was supported in part by the French National Agency ANR-10-BLANC-04030-POSTIT, ANR-16-CE30-0019-ELODIS2 and the H2020 ERC grant QUEST no. 637815.

Author information

Affiliations

1. Univ. Grenoble Alpes, CEA, INAC, PHELIQS, Grenoble, France

• Thomas Dubouchet
• , Marc Sanquer
•  & Claude Chapelier
2. Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, Grenoble, France

• Benjamin Sacépé
•  & Johanna Seidemann

• Dan Shahar

Contributions

B.S. and J.S. prepared the samples. T.D. and C.C. performed the experiments. T.D., B.S. and C.C. carried out the analysis and interpretation of the results. B.S. wrote the manuscript with the input of all co-authors. C.C. conceived and supervised the project.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to Benjamin Sacépé or Claude Chapelier.

Supplementary information

1. Supplementary Information

Supplementary Figures 1–11; Supplementary Table 1; Supplementary References 1–6