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

We study Josephson effect of $d$-wave superconductor (DS)/ferromagnet insulator(FI)/DS junctions on a surface of topological insulator (TI). We calculate Josephson current $I\left(\phi \right)$ for various orientations of the junctions where $\phi$ is the macroscopic phase difference between two DSs. In certain configurations, we find anomalous current-phase relation $I(\phi )=-I(-\phi +\pi )$ with $2\pi$ periodicity. In the case where the first order Josephson coupling is absent without magnetization in FI, $I(\phi )$ can be proportional to $\cos \phi$. 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 $I(\phi =0)\mathrm{\ne }0$. This can be used to probe the induced $d$-wave component of pair potential on TI surface in high-$T_c$ cuperate/TI hybrid structures.