Microwave Spectroscopy Evidence of Superconducting Pairing in the Magnetic-Field-Induced Metallic State of ${\mathrm{InO}}_x$ Films at Zero Temperature

We investigate the field-tuned quantum phase transition in a 2D low-disorder amorphous ${\mathrm{InO}}_x$ film in the frequency range of 0.05 to 16 GHz employing microwave spectroscopy. In the zero-temperature limit, the ac data are consistent with a scenario where this transition is from a superconductor to a metal instead of a direct transition to an insulator. The intervening metallic phase is unusual with a small but finite resistance that is much smaller than the normal state sheet resistance at the lowest measured temperatures. Moreover, it exhibits a superconducting response on short length and time scales while global superconductivity is destroyed. We present evidence that the true quantum critical point of this 2D superconductor metal transition is located at a field $B_{\mathrm{sm}}$ far below the conventionally defined critical field $B_{\mathrm{cross}}$ where different isotherms of magnetoresistance cross each other. The superfluid stiffness in the low-frequency limit and the superconducting fluctuation frequency from opposite sides of the transition both vanish at $B\approx B_{\mathrm{sm}}$. The lack of evidence for finite-frequency superfluid stiffness surviving $B_{\mathrm{cross}}$ signifies that $B_{\mathrm{cross}}$ is a crossover above which superconducting fluctuations make a vanishing contribution to dc and ac measurements.

Figure 1

(a) Temperature dependence of the sheet resistance $R_{\square }$ at different fields as indicated by the color legend. (b) $R_{\square }$ as a function of field at six fixed temperatures as shown by the color legend. The crossing point of the two lowest-temperature isotherms is approximately 7.5 Tesla.

Figure 2

Frequency dependence of the (a) real ($G_1$) and (b) imaginary ($G_2$) conductances, respectively, in the ranges $\omega /2\pi =0.08–16\mathrm{GHz}$ at the base temperature for each field. $G_1$ and $G_2$ have the same color legend at finite fields, except that $G_1$ at zero field is not plotted. The dashed grey line in (b) is a guide to the eye for $G_2\propto 1/\omega$. Arrows in (b) mark the frequencies of the maxima in $G_2$.

Figure 3

(a) Frequency dependence of the superfluid stiffness in the ranges $\omega /2\pi =0.08–16\mathrm{GHz}$ at the base temperature for each field. The color legend for fields is the same as in Fig. 2a. (b) Superfluid stiffness as a function of field at different frequencies at the base temperature for each field.

Figure 4

(a) Temperature dependence of $\Omega$ at different fields. (b) Contour plot of $\Omega$ in temperature and field. Color indicates the magnitude of interpolated values of $\Omega$ from the fitted data. (c) A phase diagram of all the quantities converted to units of Kelvin. This phase diagram can be extrapolated to $T=0$ since most of the quantities in the phase diagram saturate at low temperatures. The dashed vertical black lines in (b) and (c) mark $B_{\mathrm{cross}}$.