Theories of Superconducting Diode Effects

TL;DR

This review surveys the mechanisms behind superconducting diode effects (SDE) and Josephson diode effects (JDE), emphasizing the necessity of simultaneous TRS and IS breaking and the essential role of interference between multiple current-carrying channels. It distinguishes intrinsic bulk SDE, often realized in noncentrosymmetric superconductors via Rashba–Zeeman physics, from extrinsic vortex- and geometry-driven effects, and highlights the need for higher-order gradient terms or finite-momentum pairing to generate a diode response. The article details both phenomenological Ginzburg–Landau descriptions and microscopic treatments (Bogolyubov–de Gennes, Eilenberger) across diverse platforms, including multi-terminal JJs, vortices, FFLO/PDW states, altermagnets, and topological or valley-polarized systems. It concludes that while intrinsic mechanisms can yield sizeable diode efficiencies, robust experimental realization and device applications will benefit from exploring new symmetry-breaking routes, nonequilibrium control, and topological contexts that can enhance or reveal diode behavior in superconductors.

Abstract

Superconducting diode effects (SDE), both in bulk superconductors and in Josephson junctions, have garnered a lot of attention due to potential applications in classical and quantum computing, as well as superconducting sensors. Here we review various mechanisms that have been theoretically proposed for their realization. We first provide a brief historical overview and discuss the basic but subtle phenomenological Ginzburg-Landau theory of SDE, emphasizing the need to the simultaneous breaking of time-reversal and inversion symmetries. We then proceed to more microscopic treatments, focusing especially on implementations in noncentrosymmetric materials described by the Rashba-Zeeman model. Finally, we review proposals based on other condensed matter systems such as altermagnets, valley polarized and topological materials, and systems out of equilibrium.

Figures (5)

• Figure 1: JJ energies (top, blue) and CPRs (bottom, orange): (a) with a single harmonic $J_1$, without either AJE or JDE; (b) with a an additional first harmonic $I_1$ with ISB and TRSB resulting in a single harmonic CPR shifted by $\varphi_0\neq0$ (i.e., AJE), but no JDE; and (c) with an additional second harmonic $J_2$ resulting in JDE.
• Figure 2: Condensation energies $f(q)$ (top, blue) and supercurrents $J( q)$ (bottom, orange) as functions of the Cooper pair momentum $q$: (a) without anomalous terms and no FFLO or SDE; (b) with the TRS and IS breaking (linear) Lifshitz invariant resulting in a shift of the condensation energy by $q_0$ and a ground state with finite momentum pairing $q_0\neq0$, but no SDE; and (c) with additional anomalous terms yielding the bulk SDE. Note the similarity to Fig. \ref{['JDEfig']}.
• Figure 3: Typical condensation energy (blue) and supercurrent (orange) as functions of $q$ with ISB or TRSB: as $\alpha_2$ decreases (from left to right) and becomes negative, the minimum at $q=0$ splits into two minima at $\pm q_0$. The resulting FFLO ground state spontaneously breaks TRS and/or IS.
• Figure 4: (a) Fermi surfaces in the normal state of the Rashba-Zeeman Hamiltonian formed by the two helical bands $\lambda=\pm$ in the presence of an in-plane magnetic field (spin orientation shown by arrows). The centers of the two Fermi surfaces are shifted by $\mathbf{Q}_\lambda$ in opposite directions perpendicular to $\mathbf{h}$. (b) Schematic SC phase diagram of the Rashba-Zeeman model. Color scale shows the value of the Cooper pair momentum $q$. Black solid line indicates the second order phase transition into the normal state (white); the dashed line indicates the Lifshitz transition (crossover) between the weak and strong helical phases (with $q\sim\varsigma h$ and $q\sim h$, respectively), marked by the formation of the BFS in the strong helical phase. For some parameter values the crossover becomes a first order phase transition (solid blue line) at a critical end point (blue dot) at low temperatures. The stripe phase with coexisting $\mathbf{q}=2\mathbf{Q}_\lambda$ may occur at higher magnetic fields. (c) The BdG spectrum in the weak (bottom) and strong (top) helical phases; pink surface indicates zero energy (units are arbitrary). The spectrum is gapless in the strong helical phase as the BdG bands cross zero energy, forming the BFS (highlighted).
• Figure 5: Nonequilibrium effects in JJs. (a) Hysteresis and non-reciprocal retrapping currents in an IVC illustrating pseudo-JDE; adopted from Ref. SteinerVonOppen23. (b) Shapiro JDE in a JJ driven by AC current with amplitude $J_{AC}$, resulting in non-reciprocal Shapiro steps in DC voltage $\bar{V}$; adopted from Ref. FominovMikhailov22. (c) Multiterminal Andreev interferometer proposed for realizing proper JDE from non-equilibrium Andreev scattering processes (indicated in purple); adopted from Ref. ShafferLiLevchenko25. (d) Skewed Fraunhofer pattern measured in a multiterminal JJ with three SC terminals and a common normal region, adopted from Ref. GuptaPribiag23.