Gate-tuned superfluid density at the superconducting LaAlO${}_3$/SrTiO${}_3$ interface

The interface between the insulating oxides LaAlO${}_3$ and SrTiO${}_3$ exhibits a superconducting two-dimensional electron system that can be modulated by a gate voltage. While the conductivity has been probed extensively and gating of the superconducting critical temperature has been demonstrated, the question as to whether, and if so how, the gate tunes the superfluid density and superconducting order parameter needs to be answered. We present local magnetic susceptibility, related to the superfluid density, as a function of temperature, gate voltage, and location. We show that the temperature dependence of the superfluid density at different gate voltages collapses to a single curve that is characteristic of a full superconducting gap. Further, we show that the dipole moments observed in this system are not modulated by the gate voltage.

Figure 1

(a) The critical temperature as a function of gate voltage forms a dome. The dashed line represents our lowest measurement temperature. (b) The superfluid density at our lowest temperature as a function of gate voltage. The superfluid density increases monotonically throughout the dome. The color scale represents gate voltage and is repeated in Fig. 2.

Figure 2

(a) Superfluid density vs temperature for $V_g=110$ V, the peak of the superconducting dome. The gray area shows a systematic error. (b) Superfluid density vs temperature for every gate voltage. The colors represent the same $V_g$ from Fig. 1. (c) Normalized curves from (b). The gray line shows the temperature dependence of a weakly interacting clean BCS $s$-wave superconductor ($\Delta =1.76$ and $a=1$). The black dashed line is a fit to the data ($\Delta =2.2$ and $a=1.4$).

Figure 3

Critical temperature vs the superfluid density at lowest temperature ($T\sim 40$ mK). The red points are the data from Fig. 1 and the gray dots represent additional data sets. The dotted line is the theoretical phase fluctuation temperature from Ref. 32, which may be limiting the critical temperature on the underdoped side of the dome. The bimodal distribution on the overdoped side is due to inhomogeneity that locally suppresses $n_s$ in different regions of the sample while the $T_c$ remains the same. See also Fig. 4.

Figure 4

Susceptometry (left) and magnetometry (right) at 80 mK at different gate voltages. Inset: Reproduction of the $T_c$ dome from Fig. 1 showing the relative location of $V_g$ in each panel. (a), (b) The sample is no longer superconducting and has a paramagnetic response. Individual ferromagnetic dipoles are also visible in the paramagnetic image. (c), (d) Superconductivity appears and the landscape is relatively inhomogeneous. (e), (f) Peak of the superconducting dome, most inhomogeneity disappears. (g), (h) Excess inhomogeneity returns on the overdoped side of the dome. The ring shapes in the inhomogeneous response match the dimensions of the SQUID's field coil and are likely due to disperse pointlike defects which interact strongly with the field coil. The ferromagnetic patches do not change with $V_g$ and remain when superconductivity is gone.