High efficiency superconducting diode effect in a gate-tunable double-loop SQUID

TL;DR

This work tackles the challenge of achieving a high-efficiency superconducting diode effect (SDE) in SQUIDs by engineering non-sinusoidal current-phase relationships (CPRs) through gate-tunable Josephson junctions. The authors model a gate-tunable double-loop SQUID with three branches (each containing two JJs in series) and show that the total CPR is a sum of three branch CPRs with controllable harmonic content via gate-tuned $E_J$ values; flux sets the phase offsets $δφ_i$. Using a Monte Carlo optimization and experiments, they demonstrate a flux-tunable diode with $η$ exceeding 50% (up to about 54%) in optimized configurations, in agreement with CPR-based simulations. The results establish a pathway to higher-efficiency superconducting diodes by independent control of CPR harmonics, with potential implications for CPR-engineered qubits and lossless superconducting electronics.

Abstract

In superconducting quantum interference devices (SQUIDs), the superconducting diode effect may be generated by interference of multiple harmonic components in the current-phase relationships (CPRs) of different branches forming SQUID loops. Through the inclusion of two gate-tunable Josephson junctions in series in each interference branch of a double-loop SQUID, we demonstrate independent control over both the harmonic content and the amplitude of three interfering CPRs, facilitating significant improvement in the maximum diode efficiency. Through optimized gate-controlled tuning of individual Josephson energies, diode efficiency exceeding 50% is demonstrated. Flux-dependent oscillations show quantitative agreement with a simple model of SQUID operation.

Figures (7)

• Figure 1: (a) False-color scanning electron microscope image of a reference device. Two JJs are formed in each of the three branches (blue) of the double-loop SQUID. Electrostatic gates (gold) are then used to tune $E_J$ independently for each junction. Gold loops at the top and bottom of the device may be operated as flux lines to control flux through each loop of the SQUID independently. The black box highlights the region imaged in Fig. \ref{['fig:fig1']}(c). (b) Device schematic showing three parallel superconducting branches, each with two gate-tunable JJs in series. The device is flux-biased using an external magnetic field $B_{\perp}$. Both loops of the device are designed to have the same area. (c) False-color scanning electron microscope image of both JJs on the bottom branch of a reference device.
• Figure 2: (a) Differential resistance ($dV/dI$) plotted as a function of gate voltage $V_2$ and DC current bias $I_{bias}$. Gates $V_4$ and $V_6$ are set to $-1.1$V to pinch off the middle and bottom branches, respectively. Black arrow denotes $I_{bias}$ sweep direction for the plotted data. (b) Same as Fig. \ref{['fig:fig2']}(a), with top and bottom branches pinched off and gate $V_4$ swept. The color scale applies to both Fig. \ref{['fig:fig2']}(a) and Fig. \ref{['fig:fig2']}(b). (c) Differential resistance ($dV/dI$) as a function of $I_{bias}$ and perpendicular magnetic field $B_{\perp}$ with all junction gates set to $0$V. White arrows denote $I_{bias}$ sweep directions, with only switching current sweeps plotted. Line cut used to generate IV curve in Fig. \ref{['fig:fig4']}(d) denoted by vertical dotted white line.
• Figure 3: (a) Differential resistance ($dV/dI$) as a function of $I_{bias}$ and $B_{\perp}$ with junction gates tuned to a configuration to enhance $\eta$ ($V_2=-1.04$ V, $V_4=-1.02$ V, $V_5=-1.013$ V, $V_6=-1.005$ V). Dashed red box denotes $I_{bias}$ and $B_{\perp}$ ranges displayed in Fig. \ref{['fig:fig4']}(a). (b) $I_c^+$ (black) and $I_c^-$ (red) as a function of $B_{\perp}$ extracted from the data plotted in Fig. \ref{['fig:fig3']}(a) using a threshold resistance of $10~\Omega$. (c) Modeled $I_c^+$ and $I_c^-$ as a function of external magnetic flux threading both loops, where $\Phi_1=\Phi_2$. Calculated from the maximum and minimum values of the device CPR given by Eq. (\ref{['eq:eq3']}) using $E_{Ji}$'s displayed in Table \ref{['tab:table1']}. (d) Measured diode efficiency $\eta$ as a function of $B_{\perp}$ calculated from the switching currents plotted in Fig. \ref{['fig:fig3']}(b). (e) $\eta$ as a function of external flux ($\Phi_1=\Phi_2$) calculated from the modeled switching currents in Fig. \ref{['fig:fig3']}(c).
• Figure 4: (a) Differential resistance ($dV/dI$) for the narrow range of $I_{bias}$ and $B_{\perp}$ values denoted by the dotted box in Fig. \ref{['fig:fig3']}(a). Note that this is the same data from Fig. \ref{['fig:fig3']}(a) plotted using a different colormap to highlight smooth SC transitions. Dotted pink and gray lines denote $dV/dI$ data demonstrating high $|\eta|$ in both polarities, plotted in Fig. \ref{['fig:fig4']}(b) and Fig. \ref{['fig:fig4']}(c), respectively. (b) $dV/dI$ as a function of $I_{bias}$ at external field $B_{\perp} \sim 5.56$ μT. $\eta = -54\%$ calculated for a threshold resistance of 10 $\Omega$. (c) $dV/dI$ as a function of $I_{bias}$ at external field $B_{\perp}\sim 9.57$ μT. $\eta = 47\%$ calculated for a threshold resistance of 10 $\Omega$. (d) DC IV for device configuration with all junction gates set to $0$ V and external magnetic field set to 0 μT. Arrows at the bottom of the figure indicate $I_{bias}$ sweep direction. (e) DC IV generated from data in Fig. \ref{['fig:fig4']}(b). (f) DC IV generated from data in Fig. \ref{['fig:fig4']}(c).
• Figure 5: Demonstration of single-loop and double-loop SQUID configurations optimizing $\eta$. (a) Schematic of device expected to have the CPR given by Eq. (\ref{['eq:eqS4']}). (b) Modeled $I_c^+$ and $I_c^-$ as a function of external magnetic flux $\Phi$ for the single-loop device shown in Fig. \ref{['fig:figS3']}(a). Calculated from device CPR given by Eq. (\ref{['eq:eqS4']}). (c) $\eta$ as a function of external flux $\Phi$ calculated from the modeled switching currents in Fig. \ref{['fig:figS3']}(b). (d) CPR $I(\phi)$ in Eq. (\ref{['eq:eqS4']}) plotted for $\delta \phi=\frac{5\pi}{4}$. $I_c^+=\frac{3eE^*}{\hbar}$, $|I_c^-|=\frac{eE^*}{\hbar}$, and $\eta=50\%$. (e) Schematic of double-loop SQUID configuration expected to maximize $\eta$. The size of each 'X' corresponds to the relative $E_J$ for each junction. (f) Modeled $I_c^+$ and $I_c^-$ as a function of external magnetic flux threading both loops, where $\Phi_1=\Phi_2$. Calculated from device CPR given by Eq. (\ref{['eq:eq3']}) using $E_{Ji}$'s displayed in Table \ref{['tab:tableS2']}. (g) $\eta$ as a function of external flux ($\Phi_1=\Phi_2$) calculated from the modeled switching currents in Fig. \ref{['fig:figS3']}(f). (h) Device CPR for double-loop SQUID configuration where $\eta$ is approximately maximized. $I_c^+=95.03$ nA, $|I_c^-|=21.36$ nA, and $\eta \sim 63.3\%$.