Response tensor for the superconducting (Josephson) diode effect

Abstract

We propose a response tensor $\mathbf{\hat \chi}$ to characterize the non-reciprocal critical current response of the superconducting (Josephson) diode effect. It describes the coupling between the dipole component of the angular distribution of the critical current and the applied magnetic field -- an analogue to the Hall response in the normal state. In quasi-2D systems with Rashba spin-orbit coupling and point group symmetries $C_{3v}$, $C_{4v}$ or $C_{6v}$, this tensor takes a fully antisymmetric form. When nematicity is present, a symmetric contribution emerges, providing an indicator of the nematic order in the superconducting state. In contrast, for systems exhibiting Dresselhaus spin-orbit coupling with the $D_{2d}$ symmetry, the tensor becomes diagonal traceless, and nematicity brings in a trace part. Our analysis not only accounts for the superconducting diode effect under external applied or intrinsic effective magnetic fields, but also predicts the symmetry conditions for realizing the diode effect when the magnetic field is aligned with the current. Beyond this, the proposed tensor provides a promising tool for detecting nematicity and potential nematic transitions deep within the superconducting phase. It may also encode additional information about the underlying electronic structure and symmetry-breaking orders, warranting further experimental investigation.