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 pairs [1]. 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, T_c. A different scenario occurs in some disordered, amorphous, superconductors: Instead of a pairing-driven transition, in-coherent Cooper pairs first pre-form above T_c, causing the opening of a pseudogap, and then, at T_c, condense into the phase coherent superconducting state [2–11]. Such a two-step scenario implies the existence of a new energy scale, Δ_c, driving the collective superconducting transition of the preformed pairs [2–6]. Here we unveil this energy scale by means of Andreev spectroscopy [5, 12] in superconducting thin films of amorphous indium oxide. We observe two Andreev conductance peaks at ± Δ_c that develop only below T_c 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.

While a small amount of non-magnetic disorder weakly affects superconductivity [13, 14], the situation changes radically when disorder reaches a level such that localization of electronic wave functions 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 insulator [15, 16].

Recently a body of tunneling experiments performed on different disordered thin films [7–11, 17–19] have shown that the spectral properties of superconductivity bordering the transition to insulator deviate significantly from the Bardeen-Cooper-Schrieffer (BCS) theory [1]. The new picture drawn by these measurements is that electrons pre-form into Cooper pairs [10] well above the transition temperature, T_c, leading to a deep suppression of the single-particle density of states (DOS) at the Fermi level, which is known as the pseudogap anomaly [8–10]. Only at a temperature T = T_c at which resistivity vanishes and global phase coherence between these preformed pairs sets in, the pseudogap evolves into a hard gap, E_g. The systematic observation of an anomalously large and spatially fluctuating spectral gap E_g (see ref. [7–11]), with a ratio E_g/k_BT_c reaching up to approximately 6 in the most disordered samples [10], has pointed out a dissociation between T_c and E_g, in stark contrast to BCS superconductors for which E_g/k_BT_c = 1.76 (k_B is the Boltzmann constant).

On these grounds theory predicts that a second energy scale, Δ_c, may be at play to drive the collective, BCS-type, superconducting transition of the preformed pairs [2–6]. This collective gap is expected to scale with T_c and thus lies within the single-particle gap E_g, making it impossible to detect in a tunnelling measurement.

As earlier proposed in the context of high T_c superconductors [12, 20], the means to unveil Δ_c 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 gap [21]. 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 E_g but instead with Δ_c [5]. Consequently, whereas single-particle tunneling probes only E_g, two-particle spectroscopy, termed Andreev spectroscopy, enables direct measurement of the collective energy gap Δ_c.

Here we report on Andreev spectroscopy performed on superconducting thin films of amorphous indium oxide (a:InO) on the verge of the insulating state [22]. The transfer of electron pairs being a second order process as compared to single electron tunneling, Andreev spectroscopy requires high contact transparency. We therefore used the metallic tip of a scanning tunneling 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 tunneling to Andreev spectroscopy [23].

The chief result of this work is presented in figure 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 bottom to 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).

Fig. 1. From tunneling to Andreev spectroscopy.

Evolution of the local differential conductance G = dI/dV versus bias voltage measured on sample InO-2 at T = 0.065 K and at the same position for different values of the point-contact conductance (G(V = –3 mV)). The conductance curves are normalized to 2e²/h and have not been vertically shifted

We begin by noting that for the low point-contact conductance (botton curve in Fig 1) G(V) resembles that of single-particle tunneling with a gap E_g at the Fermi level bordered by single-particle peaks (see Fig. S2 in SI). Increasing further the conductance profoundly modifies the spectrum. While the single-particle peaks remains, 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, Andreev process at the interface leads to a monotonous increase of conductance inside the gap ultimately reaching, for perfect transparency, twice the contact conductance [24]. Our results on a highly disordered film displayed in figure 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 Δ_c 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 Figure 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 ≳ 0.6 T_c and merges farther into a conductance maximum at zero bias. This maximum, still just visible at T = T_c, disappears beyond T ≃ 1.05 T_c. 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 Fig. S8 in SI).

Fig. 2. Temperature dependence of the Andreev spectroscopy.

a, Local differential conductance curves in unit of 2e²/h measured versus bias voltage at the same position and at different temperatures. All curves except the bottom one are vertically shifted for clarity. The shift between two consecutive curves is equal to 0.12 (2e²/h). The black dashed line corresponds to the spectrum measured at T_c. b, Colour map of −d²G/dV² (arbitrary unit) versus bias voltage and reduced temperature T/T_c for the whole set of data.

The termination of the Andreev peaks at T ≃ T_c that we observe in all our samples clearly indicates that they relate to coherent superconductivity [5]. This is in stark contrast with the T evolution of the single-particle tunneling gap E_g, which remains unchanged at T_c, and only vanishes further at several times T_c, resulting in the appearance of the pseudogap in the single-particle density of states [10].

From a theoretical standpoint, the body of work on disordered superconductivity has reached a consensus in explaining the pseudogap by the pre-formation of Cooper pairs above T_c [4–6, 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, E_g, is predicted to embody two contributions. The first is the pairing energy gap Δ_p for the pre-formation 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 Δ_c related to the BCS-type condensation of the preformed pairs into the superconducting state [5]. As Δ_p is not involved when transferring two electrons simultaneously, Δ_c only can give rise to the Andreev conductance signal that we observe in our data.

Furthermore, we investigate how the values of E_g and Δ_c spreads over three highly disordered samples (samples InO-1, InO-2 and InO-3). All our measurements of E_g and Δ_c performed on these samples are summarized in Figure 3. The value of the local gap E_g obtained by fitting the density of states of spectra in the tunneling regime (see Fig. S2 and S3 in SI) is strongly dependent on the position of the tip with an an energy varying by a factor of more than three, that is, spreading between 200 µeV and 650 µeV, as previously reported [10]. In contrast, Δ_c scatters around an average value of 145 µeV with a variability of ±30 µeV that stems mainly from the experimental accuracy. Notice that this observation rules out multiple Andreev reflections at sub-multiple energies 2E_g/n, n being an integer, as a possible origin for the observed sub-gap structure in our spectra [26, 27]. Importantly, this average value of Δ_c that translates to 1.7 ± 0.3 K is close to T_c in all three samples (1.45, 1.5 and 1.9 K, see Table S1 in SI).

Fig. 3. Collective gap versus spectral gap.

Collective energy gap Δ_c as a function of single particle gap E_g measured at low temperature T ≃ 0.05 K) in contact and tunneling regimes, respectively. Δ_c and E_g are extracted from different tunneling to Andreev spectroscopy evolutions similar to those displayed in Fig. 1 and in Supplementary Fig. S3. Spectroscopy was performed at different locations on three samples InO-1, InO-2 and InO-3, identified by the red, grey and blue colors, respectively. The error bars on E_g correspond to the uncertainty in the BCS fit of the tunneling density of states. The error bars on Δ_c indicate the experimental accuracy in determining the position of the Andreev peaks.

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

Fig. 4. Two-step transition to superconductor.

Resistance in unit of h/4e² versus T normalized to T_c illustrating the typical superconducting transition of a nearly critical thin films in the vicinity of the superconductor-insulator transition. The typical distribution of Δ_c and E_g are indicated by the black arrows.

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 a:InO film (sample InO-4) that behaves in all points as a conventional dirty superconductor [28], that is, without a pseudogap above T_c (see Fig. S10b in SI). The corresponding Andreev spectroscopy shows a standard, monotonous increase of the conductance within the single-particle gap, without any intra-gap peak (see Fig. S11 in SI). This evolution, which has been observed for any position of the STM tip, can be accurately described by Blonder-Tinkham-Klapwijk (BTK) theory for N/S interfaces [24].

To conclude, the scenario drawn by our observations contrasts with standard BCS superconductors. The second energy scale we uncover together with the pseudogap comply with a two-step transition to superconductor, which develops, first, far above T_c 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

Tunneling 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 – 1000 kΩ and thus explored continuously the whole transition from tunneling to contact regime. The differential conductance $G\left(V\right)=\frac{dI}{dV}$ of the junction was measured with a lock-in amplifier technique with a voltage modulation of 10 – 30 µV.