Spontaneous supercurrents and vortex depinning in two-dimensional arrays of $\varphi_0$-junctions

TL;DR

The work demonstrates nonreciprocal vortex depinning in 2D arrays of $\varphi_0$-junctions under in-plane magnetic fields, arising from spontaneous ground-state currents induced by next-nearest-neighbor couplings. A minimal theoretical model with both nearest ($E_J$) and diagonal ($E_D$) Josephson couplings shows that phase offsets $\Delta\varphi_0$ generate a ratchet-like vortex pinning potential, yielding a magnetochiral diode effect in vortex motion. The authors observe enhanced nonreciprocity at commensurate frustration and a sign reversal near $f\approx 1/3$, with a symmetry between vortices and antivortices that distinguishes this mechanism from real-space ratchets. The results highlight the potential for engineered magnetic structures in artificial crystals lacking time-reversal symmetry and provide a framework for tuning diode-like behavior via gate voltage, field orientation, and diagonal coupling. Experimental demonstrations are supported by detailed device architectures and comprehensive transport measurements, including DC and ac (first and second harmonic) responses.

Abstract

Two-dimensional arrays of ballistic Josephson junctions are important as model systems for synthetic quantum materials. Here, we investigate arrays of multiterminal junctions which exhibit a phase difference $\varphi_0$ at zero current. When applying an in-plane magnetic field we observe nonreciprocal vortex depinning currents. We explain this effect in terms of a ratchet-like pinning potential, which is induced by spontaneous supercurrent loops. Supercurrent loops arise in multiterminal $\varphi_0$-junction arrays as a consequence of next-nearest neighbor Josephson coupling. Tuning the density of vortices to commensurate values of the frustration parameter results in an enhancement of the ratchet effect. In addition, we find a surprising sign reversal of the ratchet effect near frustration 1/3. Our work calls for the search for novel magnetic structures in artificial crystals in the absence of time-reversal symmetry.

Figures (17)

• Figure 1: Nonreciprocal transport in JJAs.a, Two-dimensional square Josephson junction array (JJA) with surface plot showing the vortex pinning potential $U(x,y)$, the bias current $I$, and the applied in-plane magnetic field $B_\text{ip}$. The JJA is modeled using both (red, orange) neighbor and next-nearest (green) neighbor Josephson couplings between superconducting islands. The lattice constant of the array is $a = 500$ nm. b, Current-voltage characteristics of the 2D JJA for different magnitude of the in-plane magnetic field. The current is always swept starting at zero absolute value. For the negative branches of the $V(I)$ curves (dotted lines) we plot the absolute values of current and voltage. The out-of-plane magnetic field is close to zero and the temperature is $T\sim 40$ mK. c, Current-voltage characteristics for different temperatures, measured at an in-plane field $B_y = 250$ mT. d, Sketch of vortex configuration at nominally zero out-of-plane magnetic field (left) and positive out-of-plane magnetic field (right). e, Rectification efficiency $\eta(B_z)$ for $\theta = -90^\circ$ and gate voltage $V_g = 0.5$ V, for different values of $B_\text{ip}$. f, Average of $\eta$ (labeled as $\bar{\eta}$) in the range $|B_z| < 20$$\mu$T as a function of $B_\text{ip}$, extracted from the data shown in panel. g, Temperature dependence of $\bar{\eta}$ for $B_\text{ip} = 250$ mT, $\theta = -90^\circ$ and gate voltage $V_g = 0.5$ V. For $T > 400$ mK there is no well-defined jump in the $V(I)$ curves which can be used to define a critical current. Therefore, we use a threshold $V_c = 30$$\mu$V to define the critical current.
• Figure 2: Tunability of rectification.a-c, Rectification efficiency $\eta = \Delta I_\text{c} / I_\text{c,mean}$ for different orientations (black arrows) of the in-plane field, measured at $B_\text{ip} = 250$ mT and temperature $T \sim 40$ mK kept constant during the entire measurement. The color corresponds to different values of the gate voltage $V_g$, varied in steps of 0.5 V from -2.5 V to 0.5 V. d, Average of $\eta$ (labeled as $\bar{\eta}$) in the range $|B_z| < 20$$\mu$T at $B_{\text{ip}} = 250$ mT as a function of gate voltage for different orientations of the in-plane magnetic field.
• Figure 3: Model of the ratchet effect.a, JJA with nearest neighbor coupling only, with no spontaneous supercurrent in the ground state. b, Sketch of the Josephson currents for a vortex in absence of diagonal coupling. The Josephson current distribution remains four-fold symmetric. A color plot of the vortex pinning potential is shown on the right. Vortices driven in the $y$-direction experience a sinusoidal pinning potential $U(y)$, as sketched above the color plot of $U(x,y)$. c, Anomalous phase shifts $\varphi_0^x > \varphi_0^{\text{diag}}$ and persistent currents for the case with diagonal coupling. d, Sketch of the Josephson currents for a vortex in the case with diagonal coupling. The corresponding vortex pinning potential $U(x,y)$ is skewed. Vortices driven in the $y$-direction experience a ratchet-like potential $U(y)$, as sketched above the color plot of $U(x,y)$ [$U(y)$ shown with exaggerated skewness for better visibility]. e, Numerical simulation of vortex depinning currents. The phase shift parameter $\Delta \varphi_0$ is the difference between horizontal and diagonal phase shifts: $\Delta \varphi_0 = \varphi_0^x - \varphi_0^{\text{diag}}$. $E_D/E_J$ is the ratio between diagonal and non-diagonal Josephson couplings. f, Resulting rectification efficiency $\eta = \Delta I_c / \langle I_{c} \rangle$ extracted from the numerical simulation of positive and negative depinning currents. The diagonal coupling is changed in steps of $0.1$ from $0$ to $0.8$.
• Figure 4: Ratchet effect at larger frustrationa-c, First harmonic $R_\omega = V_\omega / I_{ac}$ and second harmonic $R_{2\omega} = V_{2\omega} / I_{ac}$ of the resistance measured as a function of frustration and ac bias current. The in-plane field $B_\text{ip} = 125$ mT is applied perpendicular to the direction of current. d-f, Rectification efficiency $\eta$ around commensurate fields $f = 1/3$, $f = 1/2$, and $f=1$ obtained from standard $V(I)$ transport measurements. In all plots $T \sim 40$ mK.
• Figure S1: Model Hamiltonian with $\varphi_0$-junctions and diagonal couplings. (a) Phase configuration inside of a plaquette. Due to the diagonal couplings and the phase offsets of the junctions, the system is effectively subject to a transverse magnetic flux $\Delta\varphi_0 = \varphi^x_0-\varphi^\mathrm{diag}_0$ in the blue region and $-\Delta\varphi_0$ in the red region. (b) Illustration of the ground-state calculation. Only an elementary cell involving four superconducting islands need to be considered due to translational symmetry in the $x$- and $y$-directions. (c) Current configuration of the ground state. The arrows indicate the currents between the respective nodes. The length of the arrows indicates the magnitude of the current, with the critical current corresponding to an arrow connecting the nodes. Parameters: $E_D=0.5E_J$, $\Delta\varphi_0 = \pi/2$.