Diode effect in the Fraunhofer pattern of disordered planar Josephson junctions

TL;DR

This work addresses how mirror symmetry and disorder shape the Josephson diode effect (JDE) in the Fraunhofer pattern of planar Josephson junctions. Using a scattering-matrix framework and tight-binding simulations, it shows that zero-field mirror symmetry $M_x$ forbids JDE, while breaking this symmetry—via smooth gate potentials, geometric asymmetries, or short-wavelength disorder—generates robust JDE, especially at Fraunhofer nodes. The study extends to multi-terminal and multi-loop SQUID configurations, revealing phase- and geometry-tunable diode rectification up to tens of percent and highlighting mesoscopic fluctuations as a diagnostic signature. The results provide design principles for gate- and geometry-controlled JDE devices and establish a symmetry-based lens to interpret diode effects in realistic superconducting nanostructures with disorder.

Abstract

The Josephson diode effect describes the property of a Josephson junction to have different values of the critical current for different direction of applied bias current and it is the focus of intense research thanks to the possible applications. The ubiquity of the effect experimentally reported calls for a study of the impact that disorder can have in the appearance of the effect. We study the Fraunhofer pattern of planar Josephson junctions in presence of different kinds of disorder and imperfections and we find that a junction that is {\it mirror} symmetric at zero-field forbids the diode effect and that the diode effect is typically magnified at the nodal points of the Fraunhofer pattern. The work presents a comprehensive treatment of the role of pure spatial inhomogeneity in the emergence of a diode effect in planar junctions, with an extension to the multi-terminal case and to systems of Josephson junctions connected in parallel.

Figures (9)

• Figure 1: Schematics of a planar Josephson junction. A magnetic field yields a position-dependent phase that induces a diffractive Fraunhofer pattern in the critical current. The cross-product between the current and the magnetic field determines the mirror symmetry plane. (a) two-terminal junction and (b) four-terminal junction.
• Figure 2: JDE in the Fraunhofer pattern for a planar Josephson junction with long-wavelength spatial modulations. The junction is schematically depicted in (f), in which two superconducting terminals kept at phase different $\varphi$ are contacted to a normal region of width $W$, length $L$, and pierced by a magnetic field $B$ that induces a flux $\Phi=B A$, with $A=WL$ the area of the central normal region. (a) Critical currents $I_c^\pm$ as a function of the external flux $\Phi$ resulting from the local onsite potential $\delta U_i=V_g(\tanh((x_i-W/2)/W_c)+1)/2$, for $V_g=0.1~t$, $\mu=0.3~t$, and $W_c=20~a$, shown in the top-left of the figure. (b) Rectification coefficient $\eta$ for the critical currents shown in (a). (c) Dependence of the rectification coefficient $\eta$ of (b) on the gate potential $V_g$, for different values of $V_g$. (d) Critical currents $I_c^\pm$ as a function of the external flux $\Phi$ resulting from the local onsite potential $\delta U_i=V(\tanh((y_i-L/2)/W_c)+1)/2$, shown in the top-right of the figure. (e) Rectification coefficient $\eta$ for the critical currents shown in (d). The residual value arises due to the discretization of the grid. (f) Schematics of the wide Josephson junction and square lattice of lateral dimensions $W=100 ~a$ and $L=20~a$, in terms of a microscopic unit length $a$, employed in the tight-binding model to calculate the scattering matrix. Semi-infinite leads are attached to the scattering region's top and bottom. A potential barrier of strength $U_{\rm pot}=0.2~t$ is added at the interface with the leads.
• Figure 3: JDE in the Fraunhofer pattern of geometrically asymmetric junctions. (a) Critical currents $I_c^\pm$ as a function of the external flux $\Phi$ for the trapezoidal junction shown in the top-right inset, with width $W=100~a$ and left and right lengths $L_l=25~a$, $L_r=15~a$. (b) Rectification coefficient $\eta$ for the critical currents shown in (a). (c) Critical currents $I_c^\pm$ as a function of the external flux $\Phi$ for the trapezoidal junction shown in the top-right inset, with bottom and top width $W_b=120~a$, $W_t=80~a$, and length $L=20~a$. (d) Rectification coefficient $\eta$ for the critical currents shown in (c). For all panels we set $\mu=0.3~t$, barrier $0.2~t$, and $\delta U_i=0$.
• Figure 4: Fraunhofer interference pattern of a planar Josephson junction with short-range disorder. The junction structure is as in Fig. \ref{['Fig2']}. (a) Critical currents $I_c^\pm$ as a function of the external flux $\Phi$ resulting from the local onsite potential $\delta U_i$, with $-U_0/2<\delta U_i<U_0/2$ randomly distributed for $U_0=0.1~t$, shown in the top-right inset. (b) Rectification coefficient $\eta$ for the critical currents shown in (a). (c) Critical currents $I_c^\pm$ as a function of the external flux $\Phi$ resulting from a local onsite potential $\delta U_i$, with $-U_0/2<\delta U_i<U_0/2$ randomly distributed for $U_0=0.1~t$ and symmetrized $\delta U_i\to (\delta U_i+M_x \delta U_i M_x^{-1})/2$, shown in the top-right inset. (d) Rectification coefficient $\eta$ for the critical currents shown in (c). The chemical potential is set to $\mu=0.3~t$.
• Figure 5: Statistical analysis of disordered configurations. (a) Rectification coefficient $\eta$ for 50 disorder configurations and a given value of the strength $U_0=1.2~t$. In black is the average rectification of the 50 random disorder configurations. (b) Root mean square of the rectification coefficient, ${\rm rms}~\eta\equiv \sqrt{\langle\eta^2\rangle-\langle\eta\rangle^2}$, averaged over 50 different pseudorandom disorder configurations, versus the applied flux for different values of $U_0$. The curve corresponding to $U_0=1.2~t$ is the square root of the average of the square of the curves in (a). In (a) and (b) $\mu=0.3~t$, $L=20$, $W=100$ and there is no barrier. (c) ${\rm rms}~\eta\times(t/U_0)$ providing a scaling of the curves in (b) with the impurity strength.