Nonequilibrium Second-Order Phase Transition in a Cooper-Pair Insulator

In certain disordered superconductors, upon increasing the magnetic field, superconductivity terminates with a direct transition into an insulating phase. This phase is comprised of localized Cooper pairs and is termed a Cooper-pair insulator. The current-voltage characteristics measured in this insulating phase are highly nonlinear and, at low temperatures, exhibit abrupt current jumps. Increasing the temperature diminishes the jumps until the current-voltage characteristics become continuous. We show that a direct correspondence exists between our system and systems that undergo an equilibrium, second-order, phase transition. We illustrate this correspondence by comparing our results to the van der Waals equation of state for the liquid-gas mixture. We use the similarities to identify a critical point where an out of equilibrium second-order-like phase transition occurs in our system. Approaching the critical point, we find a power-law behavior with critical exponents that characterizes the transition.

Figure 1

Discontinuity in the $I\text{−}V$ characteristics. $I$ (using a logarithmic scale) vs $V$. The data points are marked by crosses and the solid lines are a guide to the eye. At $T=120\mathrm{mK}$, a $\mathrm{\Delta }I$ of 3 orders of magnitude is apparent at $V_{\text{th}}=21\mathrm{mV}$. $\mathrm{\Delta }I$ decreases with $T$ and the $I\text{−}V$ turns continuous at $T\ge 190\mathrm{mK}$. Inset: $R$ (logarithmic scale) vs $B$. The color-coded $T$’s are (from red to purple) $T=1$, 0.6, 0.4, 0.2, and 0.1 K. The dashed black line marks $B=3\mathrm{T}$, where the $I\text{−}V$’s in the main figure were measured.

Figure 2

Resemblance between vdW and $T_{\text{el}}\text{−}V$’s isotherms. (a) ($-V$) vs $T_{\text{el}}$ isotherms of amorphous indium-oxide in the insulating phase bordering superconductivity. The vdW equivalent critical point is marked by a black circle. The $T_{\text{el}}\text{−}V$’s turn continuous at $T_{\text{ph}}=T_c=190\mathrm{mK}$. The color-coded $T_{\text{ph}}$’s are evenly separated by $\mathrm{\Delta }{T}_{\text{ph}}=10\mathrm{mK}$ starting from $T_{\text{ph}}=100\mathrm{mK}$ (purple) up to $T_{\text{ph}}=200\mathrm{mK}$ (red). (b) Solutions of the normalized vdW Eq. (1) with Maxwell’s construction on a $\widehat{P}-\widehat{v}$ diagram (see text). For proper comparison between isotherms of both figures, the value of each $\tau$ in (b) equals $T_{\text{ph}}/T_c$ of its corresponding color in (a).

Figure 3

Electron heating—Critical exponents. (a) $\mathrm{\Delta }{T}_{\text{el}}$ vs $t_{\text{ph}}\equiv (T_c-T_{\text{ph}}/T_c)$ (log-log scale) with $T_c=190\mathrm{mK}$. The data are presented as red triangles while the power law fit is marked by a dashed black line. The resulting critical exponent is $\tilde{\beta }=0.9$. (b) $T_{\text{el}}-T_c^{\text{el}}$ vs $V-V_c$ (log-log scale). The data, in blue circles, were measured along the critical isotherm. The dashed black line is the power law fit. The resulting critical exponent is $\tilde{\delta }=1.86$.

Figure 4

Phase diagram and heating time. (a) ($-V$)-$T_{\text{ph}}$ diagram. The red dots represent $V_{\text{th}}$ of each $T_{\text{ph}}$ isotherm. This plot is analogous to a vdW $P-T$ phase diagram. The dashed line corresponds to the vapor-liquid coexistence curve and terminates at the critical point. (b) $\tau$ vs $T_{\text{ph}}$. At low $T_{\text{ph}}$ the characteristic heating time is $\tau \sim 10^{-2}\sec$, close to $T_c=190\mathrm{mK}$ $\tau \sim 10^{-4}\sec$.