Superconducting diode effect in fractal superconductors: fractional-order Ginzburg-Landau theory for Josephson junctions

Abstract

We develop a fractional-order Ginzburg-Landau (GL) framework for nonreciprocal superconducting transport in Josephson junctions formed by fractal superconductors or superconducting media with nonlocal correlations, separated by a noncentrosymmetric normal layer. We show that nonreciprocity and the superconducting diode effect arise from the interplay between the Lifshitz invariant and fractional kinetics, with the latter serving as an effective, symmetry-consistent representation of fractal geometry and finite-range memory. Two complementary approaches are pursued. In a fractional integral GL formulation, spatial integration on a fractal space yields analytic solutions and reveals how rectification scales with the dimensionality of the fractal media and the strength of the Lifshitz-like drift. In a fractional derivative-based formulation derived via the Agrawal variational principle with left/right Caputo operators, we obtain a gauge-invariant free energy, the corresponding GL equations, and a current density. We use fractional orders as effective parameters that represent nonlocal and memory effects induced by fractal microstructure. Within a two-mode plane-wave approximation we derive a compact current-phase relation and an expression for the diode efficiency, and we map the rectification amplitude across the fractional kinetic and the Lifshitz/memory order. An exact single-sided solution in terms of Prabhakar functions further confirms robust, tunable nonreciprocity, including a near-ideal diode response. This identifies a pathway to near-perfect superconducting diodes by engineering fractal (fractional-kinetic) transport achieved by tuning the fractional orders and Lifshitz strength without invoking magnetic fields or geometric ratchets. In the integer limit of local kinetics and Lifshitz-like drift, both constructions reduce to the standard $\varphi_0$ Josephson junction.