Anomalous Josephson effect in d-wave superconductor junctions on Tl surface

We study the Josephson effect of d -wave superconductor (DS)/ferromagnet insulator (FI)/DS junctions on a surface of topological insulator (TI). We calculate Josephson current I(φ) for various orientations of the junctions where φ is the macroscopic phase difference between two DSs. In certain configurations, we find anomalous current-phase relation I(φ)=−I(−φ+π) with 2π periodicity. In the case where the first-order Josephson coupling is absent without magnetization in FI, I(φ) can be proportional to cosφ . In symmetric junctions, the magnitude of the obtained Josephson current is enhanced due to the zero-energy states on the edge of DS on TI. Even if we introduce an s -wave component of pair potential in DS, we can still expect the anomalous current-phase relation in asymmetric DS junctions with I(φ=0)≠0 . This can be used to probe the induced d -wave component of pair potential on a TI surface in high-T c cuprate/TI hybrid structures.