Competition between electron pairing and phase coherence in superconducting interfaces

In LaAlO3/SrTiO3 heterostructures, a gate tunable superconducting electron gas is confined in a quantum well at the interface between two insulating oxides. Remarkably, the gas coexists with both magnetism and strong Rashba spin–orbit coupling. However, both the origin of superconductivity and the nature of the transition to the normal state over the whole doping range remain elusive. Here we use resonant microwave transport to extract the superfluid stiffness and the superconducting gap energy of the LaAlO3/SrTiO3 interface as a function of carrier density. We show that the superconducting phase diagram of this system is controlled by the competition between electron pairing and phase coherence. The analysis of the superfluid density reveals that only a very small fraction of the electrons condenses into the superconducting state. We propose that this corresponds to the weak filling of high-energy dxz/dyz bands in the quantum well, more apt to host superconductivity.

by silver epoxy to ensure negligible impedance contacts. A tiny metallic contact on the MgO substrate allows the gate to be connected to an external contact pad through an Al wire-bond.    the nominal values of L 1 and C 1 within less than 5% of error. It varies by less than 3% over the entire temperature range 20 mK -300 K and by less than 0.1% below 1 K. More over, this weak variation of ω 0 with temperature is likely due to the capacitor C 1 which is replaced by the sample in the real experiment.
The value of L 1 , which is determined by the geometrical inductance of the inductor, is not expected to vary with temperature. Four points DC measurements also confirm that the resistance R 1 doesn't change significantly with temperature (< 0.5%). In any case, R 1 doesn't enter into the determination of the resonance frequency and consequently in the determination of J s .

Supplementary Note 2: Accuracy on the determination of J s
The total capacitance of the circuit is extracted from the resonance frequency in the normal state. Supplementary Figure 2b shows the temperature dependence of ω 0 of the sample circuit in the temperature range 0.4 -300 K which illustrates clearly the quantum paraelectric nature of SrTiO 3 . Above 10 K, the resonance frequency increases with temperature because of the reduction of the dielectric constant. The saturation of ω 0 above 200 K is due to the parasitic capacitance of the circuit C para 3.5 pF. Below 10 K, quantum fluctuations lead to a saturation of the dielectric constant and ω 0 becomes temperature independent. This is in particular true in the temperature range of interest (< 500 mK) as seen in Fig. 3a of the manuscript. A low temperature, the total capacitance of the circuit is mainly dominated by C STO ( 45 pF for V G = 0 V). Although it is the total capacitance of circuit that is measured in the normal state and plotted in Fig. 2c, we refer to it as C STO for sake of clarity (C STO +C para C STO ).
The determination of the superfluid stiffness J exp s relies on the measurement of the kinetic inductance L k , which is extracted from the resonance frequency ω 0 in the superconducting state. This latter depends only on the total inductance and capacitance of the sample circuit. The total resistance of the circuit is not involved in the determination of J s but it must be sufficiently close to Z 0 =50 Ω to generate a visible absorption dip. The accuracy of the method can be estimated from the uncertainty in the value of the total inductance of the circuit in the normal state. In practice, we assume that by design, L 1 dominates the total inductance of the circuit. This SMD component has a nominal value of 10 nH with a tolerance of 5%. In addition we neglected in first approximation the geometrical inductance of the 2-DEG which can be estimated with the following relation [2] L G = 0.2l[ln 2l w + t + 0.5 + 0.02235 where l is the length of the 2-DEG expressed in mm (3mm), w is its width (3 mm) and t its thickness (∼10 nm). As t w, L G is essentially independent of t and we obtain L G 0.85 nH. The error margin in the determination of J exp s , taking into account the 5% tolerance on L 1 and the contribution of the geometrical inductance, is indicated by a grey outline in Figure 4a of the main text. It is lower than 15 % in the entire phase diagram.
Supplementary Note 3: Temperature dependence of J exp s In the overdoped region where the zero-temperature stiffness coincides with the BCS prediction, the gap energy can be extracted from a fit of the temperature dependence of J exp s (T ) [3]. As seen in Supplementary Figure 6, a good agreement is obtained between experimental data at V G = 34 V and the BCS model in the low temperature part of the curve (T < T c /2) for ∆(0) = 24 µeV. This value is very close from ∆ exp s = 22.2 µeV that was obtained from the conversion of J exp s (0) into a gap energy through Eq. (2). On the other hand, at higher temperature, J exp s (T ) deviates from the BCS prediction in a Berezinskii-Kosterlitz-Thouless like jump as expected in the presence of vortex fluctuations. However, the jump occurs before the intersection with the universal line 2T π and it is smeared out. A quantitative analysis of this issue, which must include the contribution of spatial inhomogeneities [4], will require more further theoretical developments.