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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(φ\right) $ 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 $I(φ)=-I\left( -φ+π\right)$ with $2π$ periodicity. In the case where the first order Josephson coupling is absent without magnetization in FI, $I(φ)$ can be proportional to $\cos φ$. 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 $I\left( φ=0\right) \neq 0$. 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.