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 Josephson junction geometries. The first consists of two ferromagnetic layers between conventional $s$-wave superconductors, with magnetizations along the $x$- and $y$-axes, forming a bilayer junction. The second contains three ferromagnetic layers between two $s$-wave superconductors, with magnetizations along the $x$-, $y$-, and $z$-axes, forming a trilayer junction. Both systems are analyzed in the short and long junction limits. In the bilayer case, where no anomalous Josephson current is present, odd-frequency equal-spin triplet correlations develop pronounced peaks at finite magnetizations in the short junction limit for both tunneling and transparent interfaces. The odd-frequency mixed-spin triplet correlations also exhibit peaks at finite magnetizations for tunneling interfaces, whereas for transparent interfaces they display both peaks and zeros. In the trilayer case, where an anomalous Josephson current exists, similar peaks in both equal- and mixed-spin odd-frequency triplet correlations occur at finite magnetizations for tunneling and transparent interfaces. The spatial profiles of these correlations remain largely unaffected by the anomalous current. The Josephson diode efficiency is finite and reaches its maximum at magnetizations corresponding to the peaks of the anomalous current. Overall, our results show that the anomalous Josephson current has only a marginal influence on odd-frequency spin-triplet pairing. This indicates that the emergence of odd-frequency correlations and the Josephson diode effect are largely independent phenomena, contrary to earlier conjectures. Analysis of the long junction limit leads to the same qualitative conclusions for both configurations.