Marginal Influence of Anomalous Josephson Current on Odd-Frequency Spin-Triplet Pairing in Ferromagnetic Josephson Diodes

We examine how an anomalous Josephson current influences odd-frequency superconducting correlations in two distinct Josephson junction geometries. The first configuration consists of two ferromagnetic layers sandwiched between conventional s-wave superconductors, with the magnetization vectors of the ferromagnets misaligned. The second involves three ferromagnetic layers embedded between two s-wave superconductors, with their magnetizations oriented along the x-, y-, and z-axes, respectively. In the first case, where the anomalous Josephson current is absent, odd-frequency spin-triplet correlations develop pronounced peaks at finite magnetization strengths in both the tunneling and transparent limits, while the equal-spin triplet component exhibits zeros at finite magnetizations in the transparent regime. In the second configuration, where an anomalous Josephson current is present, similar peaks in odd-frequency spin-triplet pairing appear at finite magnetizations under both transport regimes, and the spatial profile of these correlations remains largely unaffected by the current\rq{}s presence. The Josephson diode efficiency is finite and attains its maximum at magnetization values corresponding to the peaks of the anomalous current. Overall, our results demonstrate that the anomalous Josephson current has only a marginal effect on odd-frequency spin-triplet pairing, suggesting that the emergence of odd-frequency correlations and the Josephson diode effect are largely independent phenomena, contrary to some earlier conjectures.