Two-dimensional (2D) electron systems are fascinating, being characterized by phenomena such as the quantum Hall effect, ferromagnetism, and superconductivity. The 2D-superconducting state yields substantial critical temperatures
despite the presence of phase and amplitude fluctuations of the order parameter. The large susceptibility to fluctuations results in many cases in a bosonic 2D superconductor–to–insulator (SIT) transition, with Cooper pairs existing in the insulating state. This transition is generally induced by tuning disorder or by changing the carrier density. The LaAlO3–SrTiO3 interface 2D electron liquid (2DEL)  is a 2D superconductor that can be driven through a SIT by depleting charge carriers with an electric field. This superconductor is of great interest because the superconducting state exists at extremely small carrier densities.
In the insulating state of the LaAlO3–SrTiO3 2DEL, Cooper pairs are present, as evidenced by the fact that the resistance at the transition equals the quantum resistance of paired electrons h/4e2. Furthermore, tunneling measurements find a superconducting gap in the density of states across the transition. Depending on their interaction, the Cooper pairs possibly form a macroscopic quantum-state characterized by an order parameter, a coherence length, and a critical magnetic field. It is also possible that the Cooper pairs act as single particles if their interaction is non-existent or weak. Which of the two scenarios occurs in a superconductor with small carrier density is unknown, yet important for the general understanding of superconductivity. Both scenarios, however, are fundamentally different, as exemplified by their response to applied perpendicular magnetic fields H. If the Cooper pairs are phase coherent, the perpendicular upper critical field Hc is determined by vortex behavior, whereas if the Cooper pairs are localized without phase coherence, Hc is determined by pair breaking due to the Zeeman energy. In the above cases, Hc differs considerably and for LaAlO3–SrTiO3 it equals 0.3 and 0.8 T, respectively. This difference opens a route to determine unequivocally the Cooper-pair nature of the insulating state, as Hc can be well measured. To measure Hc precisely, we use magnetic-field-dependent tunneling spectroscopy. In tunneling spectroscopy measurements, the disappearance of the superconducting gap in the spectra quantitatively yields Hc for the superconducting as well as for the insulating side of the SIT.
To perform the tunneling spectroscopy measurements, planar tunnel junctions were fabricated, the two electrodes of which were provided by the 2DEL and by a gold layer, respectively . Measurements with these junctions resolved the superconducting gap of the 2D state. The carrier concentration of the 2DEL was tuned electrically by a back-gate voltage VG, allowing tunnel measurements across the entire superconducting dome (maximum Tc ≈ 300 mK) as well as in the insulating state. Tunnel spectra were obtained by applying a current between the top electrode and the interface and by simultaneously recording the tunneling voltage as well as the conductance. The polarity of the voltage characterizes the sign of the interface voltage with respect to the top electrode bias. Even in the insulating state, the minimal tunneling resistance significantly exceeds the maximal resistance of the 2DEL. All measurements were taken at a temperature of ≈60 mK. To measure Hc, the disappearance of the superconducting gap was analyzed as a function of magnetic field H. Figure 1 shows tunnel spectra as a function of H with VG = 0. In these tunnel junctions, the differential conductance reflects the density of states of the 2DEL, and the superconducting gap Δ is observed with a value of ≈60 μV. The suppression of the density of states at V = 0 and the quasiparticle peaks disappear gradually with increasing H, as expected for type-II superconductors.
We now turn to the carrier-density dependence of the critical field. The dI/dV (V) spectra for four different values of gate voltage are shown in Fig. 2. The four panels cover the entire density range, from the overdoped to the metallic / insulating side. In all cases the superconducting gap is present at 0 T. The gap increases with decreasing VG, i.e., decreasing carrier concentration n.
In Fig. 3 the measured values for Hc are given across the superconducting dome and the SIT. The critical field can be readily determined from these spectra . Hc monotonically increases with decreasing charge carrier density. Starting at 80 mT in the overdoped range, Hc (VG) reaches 300 mT in the underdoped range. Also in the insulating regime (–200 and –300 V), Hc ≈300 mT. Now we compare the critical fields with those expected for individual localized Cooper pairs (Chandrasekhar-Clogston critical field Hccc) and with the critical field being caused by superconducting vortices (HcBCS). As shown in Fig. 3, the measured data for the insulating state agree only with the vortex critical field HcBCS, with a notable disagreement with the critical field expected for individual Cooper pairs Hccc. Indeed, the entire gate-voltage dependence of Hc can be well described by the model based on the existence of conventional (BCS-type) vortices.
In conclusion, tunneling spectroscopy performed on the two-dimensional superconductor generated at LaAlO3–SrTiO3 interfaces provides evidence that the electron system in the insulating state consists of one or more fluctuating ensembles of macroscopically phase-coherent Cooper pairs. This understanding is corroborated by the existence of the coherence peaks in the tunnel spectra in the insulating state. The data exclude an electron system consisting solely of superconducting puddles with a length scale smaller than the superconducting coherence length.