Survival of superconducting correlations across the two-dimensional superconductor-insulator transition: A finite-frequency study

The complex ac conductivity of thin highly disordered $\mathrm{In}{\mathrm{O}}_x$ films was studied as a function of magnetic field through the nominal two-dimensional superconductor-insulator transition. We have resolved a significant finite-frequency superfluid stiffness well into the insulating regime, giving direct evidence for quantum superconducting fluctuations around an insulating ground state and a state of matter with localized Cooper pairs. A phase diagram is established that includes the superconducting state, a transition to a “Bose” insulator, and an eventual crossover to a “Fermi” insulating state at high fields. We speculate on the consequences of these observations, their impact on our understanding of the insulating state, and its relevance as a prototype for other insulating states of matter that derive from superconductors.

Figure 1

(Color online) (a) Temperature dependence showing the broad resistive transition resulting from fluctuations. The normal state resistance curves are generated using the procedure described in the text. The amplitude temperature $T_{co}$ is believed to be $2.28\mathrm{K}$. (b) dc Sheet resistance vs temperature for field values $H=0$, 0.05, 0.1, 0.25, 0.5, 1, 1.5, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, and $8\mathrm{T}$. (c) Sheet resistance vs magnetic field for temperatures shown. The crossing point determines a critical field of $3.68\mathrm{T}$.

Figure 2

(Color online) Superfluid stiffness, $T_{\theta }$ $(\propto \omega {\sigma }_2)$, extracted as described in the text. (a) Temperature dependence at $H=0$. Also shown is the theoretical $T_{\mathit{KTB}}$ which intersects $T_{\theta }$ around $T=1.1\mathrm{K}$. (b) $T=0.5\mathrm{K}$ magnetic field dependence at select low frequencies.

Figure 3

(Color online) (a) ac Sheet resistance $[\propto \mathrm{Re}\left(\frac{1}{\sigma d}\right)]$. The bold line is the critical resistance extrapolated to finite temperature. (b) Superfluid stiffness $(\propto \omega {\sigma }_2)$ at $22\mathrm{GHz}$. The critical field $H_{\mathit{SIT}}$ is shown as a white dot at $H=3.68\mathrm{T}$. Yellow indicates maximum.

Figure 4

(Color online) 2D field-tuned superconductor-insulator “phase diagram.” Contours with markers are the detection limit for $T_{\theta }$ at specified frequencies. Black dots on the temperature axis denote $T_{\mathit{amp}}$ $(=T_{c0})$ and $T_{\mathit{\text{phase}}}$ $(=T_{\mathit{KTB}})$, signifying temperatures where the amplitude and phase, respectively, become well defined. $H_{\mathit{SIT}}$ appears as a black dot and the contour of critical resistance from Fig. 3a appears as a solid black line. Open symbols represent data obtained from a small linear extrapolation beyond our maximum field of $8\mathrm{T}$.