Theoretical study of superconducting diode effect in planar $T_{d}-MoTe_{2}$ Josephson junctions

We investigate the Josephson diode effect (JDE) within quasi-2D planar systems featuring the $C_{1v}$ spin-orbit coupling (SOC) and Zeeman fields in the normal region. Our analysis is based on experimental observations conducted on $MoTe_{2}$ planar Josephson junctions (JJ) subjected to out-of-plane magnetic fields. We emphasize the pivotal role of symmetry breaking in current directionality for the occurrence of the JDE. Specifically, we observe the emergence of asymmetric Andreev bound states (ABSs) and $0$-$\pi$-like transitions with $\varphi_0$-shifts in the current phase relations (CPRs) in systems with specific symmetry breaking induced by SOC and Zeeman fields, leading to different critical current magnitudes in opposite directions. Additionally, we explore the influence of parameters such as the strength of SOC, Zeeman field magnitude and orientation, conduction channels with different transverse momenta, and junction lengths on the JDE efficiencies. Our results indicate the potential for diverse approaches to modulate efficiencies and provide insights that can aid in the discovery of materials and design of Josephson diodes with significantly enhanced efficiency.