Bimodal Phase Diagram of the Superfluid Density in ${\mathrm{LaAlO}}_3/\mathrm{SrTi}{\mathrm{O}}_3$ Revealed by an Interfacial Waveguide Resonator

We explore the superconducting phase diagram of the two-dimensional electron system at the ${\mathrm{LaAlO}}_3/\mathrm{SrTi}{\mathrm{O}}_3$ interface by monitoring the frequencies of the cavity modes of a coplanar waveguide resonator fabricated in the interface itself. We determine the phase diagram of the superconducting transition as a function of the temperature and electrostatic gating, finding that both the superfluid density and the transition temperature follow a dome shape but that the two are not monotonically related. The ground state of this two-dimensional electron system is interpreted as a Josephson junction array, where a transition from long- to short-range order occurs as a function of the electronic doping. The synergy between correlated oxides and superconducting circuits is revealed to be a promising route to investigate these exotic compounds, complementary to standard magnetotransport measurements.

Figure 1

A coplanar waveguide resonator at the LAO/STO interface. (a) Sketch of the 2DES (magenta) at the LAO/STO (yellow/gray) interface before (left) and after (right) the lithography. The field effect is obtained by tuning the voltage ($V_{\mathrm{BG}}$) of the back gate. (b) Experimental setup to probe the SCWR. The $-65\mathrm{dBm}$ attenuation is distributed across the stages from room temperature to the mK plate. (c) Reflection spectrum of the SCWR measured at 11 mK, $V_{\mathrm{BG}}=0\mathrm{V}$, and $-95\mathrm{dBm}$.

Figure 2

Temperature dependence of the superfluid characteristics. (a) Frequencies of the cavity modes from $m=2$ (blue squares) to $m=5$ (yellow reversed triangles) at $V_{\mathrm{BG}}=0\mathrm{V}$. (b) Pearl length and (c) superfluid density calculated from the data in (a) using Eqs. (1) and (2). The solid line in (c) is the best fit of Eq. (3) for $m=3$, which gives $T_c=133\pm 2\mathrm{mK}$. (d) Resistance-current characteristics measured on the ground plane.

Figure 3

Superfluid density under the field effect. (a),(b) Reflection amplitude of the SCWR as a function of $V_{\mathrm{BG}}$ at (a) 11 and (b) 133 mK. The white dashed line is a guide to the eye highlighting $m=3$. (c) Temperature-back-gate phase diagram of the superfluid density calculated from $m=3$.

Figure 4

Analysis of the superfluid gate dependence. (a) $T_c$ and $n(11\mathrm{mK})$ as a function of $V_{\mathrm{BG}}$. $T_c$ is calculated by fitting the data from Fig. 3 with Eq. (3). (b) Critical temperature as a function of the superfluid density at 11 mK.