Diode effect in a d-wave superconductor planar Josephson junction

We consider phase-controlled planar Josephson junction comprising a two-dimensional electron gas with strong spin-orbit coupling and d-wave superconductors.

We investigate the superconducting diode effect that originates from a mirror and time-reversal symmetry breaking in the above planar Josephson junction. Josephson's current is calculated analytically and numerically for different pairing orientations using Matsubara Green's function approach. We also extend our analysis to d+id', d+is pairings realizable in twisted cuprate bilayers. The nonreciprocity is quantified by the quality factor $Q = frac{I_{c}^R-I_{c}^L}{I_c^R+I_c^L}$, where $I_c^R(I_c^L)$ refers to the maximum forward (backward) Josephson current, for different system parameters, including the Zeeman-field, spin-orbit coupling, pairing orientation, and gate bias. Furthermore, we try to differentiate between the super-current that arises due to the presence of edge states and the bulk states. We optimize the quality factor for various system parameters including the length of the junction and d-wave pairing angle. Finally, we discuss the relevance of Majorana-bound states and how they influence the observed nonreciprocity.