The Berezinskii-Kosterlitz-Thouless Transition and Anomalous Metallic Phase in a Hybrid Josephson Junction Array

Abstract

We investigate the Berezinskii-Kosterlitz-Thouless (BKT) transition in a semiconductor-superconductor two-dimensional Josephson junction array. Tuned by an electrostatic top gate, the system exhibits separate superconducting (S), anomalous metal (M*), and insulating (I) phases, bordered by separatrices of the temperature-dependent of sheet resistance, $R_{s}$. We find that the gate-dependent BKT transition temperature falls to zero at the S-M* boundary, suggesting incomplete vortex-antivortex pairing in the M* phase. In the S phase, $R_{s}$ is roughly proportional to perpendicular magnetic field at the BKT transition, as expected, while in the M* phase $R_{s}$ deviates from its zero-field value as a power-law in field with exponent close to 1/2 at low temperature. An in-plane magnetic field eliminates the M* phase, leaving a small scaling exponent at the S-I boundary, which we interpret as a remnant of the incipient M* phase.

Figures (4)

• Figure 1: Sheet resistance, $R_{s}(T)$, of the InAs/Al Josephson junction array (inset) as a function of temperature, $T$, at several gate voltages, $V_{G}$, ranging from $-3.1$ V to $-3.9$ V, from Ref. Boettcher.2018. Two separatrices (blue and orange curves) mark the boundaries between superconducting (S) phase, where $R_{s}(T)$ is concave down, becoming unmeasurably small at low $T$, the anomalous metal (M*), where $R_{s}(T)$ is concave up and saturating at low $T$, and the insulating phase, where $R_{s}(T)$ increases with falling $T$. The S-M* boundary (blue) starts at normal-state sheet resistance, $R_{N}\sim 1$ k$\Omega$. The M*-I (orange) is roughly temperature independent, with $R_{s} \sim 6$ k$\Omega$, similar to with a conventional S-I boundary. Inset: false-color electron micrograph with scale bar showing the array before the deposition of the top insulator and metallic gate, with 1 $\mu$m Al squares patterned onto an InAs substrate, as described in Ref. Boettcher.2018.
• Figure 2: (a-c) Sheet resistance, $R_{s}(T)$ as a function of temperature, $T$ at three gate voltages (filled colored circles), parameterized by normal-state sheet resistance, $R_{N}$, along with fits to Eq. \ref{['HN']}. Top axes show gate-independent critical temperature, $T_{c0}$ of the Al squares, read directly from the data, and BKT transition temperature, $T_{\rm BKT}$ from the fits. Note the good fit deep in S, which becomes poorer toward the S-M* boundary. (d) BKT transition temperature $T_{\rm BKT}$ extracted from fits at many gate voltages (black filled circles), as a function of $R_{N}^{-1}$. $T_{\rm BKT}$ reaches zero near the S-M* boundary, defined by the blue separatrix in Fig. \ref{['fig1']}. Three colored data points correspond to panels a-c. Model line is based on linear relations $T_{\rm BKT} \sim E_{J} \propto I_{c} \propto R_{N}^{-1}$, with a single fit parameter (see text).
• Figure 3: Low-field magnetoresistance $R_{s}(B_{\perp}, T)$ as a function of perpendicular magnetic field, $B_{\perp}$, shows a power-law dependence (straight line on a log-log plot), $R_{s}(B_{\perp}, T)- R_{s}(0, T) = A(T) B_{\perp}^{\beta(T)}$ with a power $\beta(T)$ dependence on temperature, $T$ (marked on graphs), and gate-controlled normal-state sheet resistance, $R_{N}$. (a) In the superconducting (S) phase, on the low-resistance (less negative $V_G$) side of the S-M* boundary, with $R_{N}=274\, \Omega$, $\beta(T_{\rm BKT}) \sim 1$. (b-c) In M*, $\beta(T)< 1$ at low $T$, and appears to settle around $\beta \sim 1/2$. Fluctuations in $R_{s}(B_{\perp})$ on a $\sim 1$ mT scale result from flux commensuration effects.
• Figure 4: (a) Sheet resistance, $R_{s}$, as a function of gate voltage, $V_{G}$ at $B_{\parallel} = B_{\perp} =0$ shows $T$-independent curves at low $T$ across the anomalous metal phase, M*, which flair out for different $T$ in the superconducting (S) and insulating (I) phases. Inset: Scaling using only high-$T$ isotherms ($T>0.2$ K) at the M*-I boundary (orange circle in main panel) yields exponent $\alpha = 0.72$. (b) Scaling using only low-$T$ isotherms ($T<0.2$ K) at the M*-I boundary yields $\alpha = 0.13$. (c) Scaling at the S-M* boundary [blue circle in (a)], yields $\alpha = 0.01$. (d) Applying an in-plane magnetic field, $B_{\parallel} = 0.5$ T, results in a single crossing of isotherms (black circle). Inset: Scaling with all isotherms yields $\alpha = 0.1$. We interpret the small $\alpha$, i.e., nearly parallel isotherms, as a vestige of M*, where low-$T$ isotherms coincide. The broad S-I transition (white region) is bounded by the crossing point on the I side and the rapid spreading of isotherms on the S side. The top axis indicates where $T_{\rm BKT} \rightarrow 0$ following similar analysis to Fig. \ref{['fig2']}. Data for Device B.