Theory of diode effect in d-wave superconductor junctions on the surface of topological insulator

Nonreciprocal responses of noncentrosymmetric quantum materials attract recent intensive interests, which is essential for the rectification function in diodes. A recent breakthrough is the discovery of superconducting diode effect. The principle to enlarge rectification effect is highly desired to guide the design of superconducting diode. Here, we study theoretically the Josephson junction S/FI/S (S: $d$-wave superconductor, FI: ferromagnetic insulator) on the surface of a topological insulator (TI). The simultaneous existence of $\sin\varphi$, $\cos\varphi$ and $\sin 2\varphi$ terms with almost the same order in Josephson current $I(\varphi)$ is essential to get larger values of $Q$ factor given by $Q=(I_{c}^{+} - \mid I_{c}^{-} \mid)/ (I_{c}^{+} + \mid I_{c}^{-} \mid)$ with $I_{c}^{+}={\rm max}(I(\varphi))$ and the negative one $I_{c}^{-}$ for macroscopic phase difference $\varphi$ of two superconductors on TI. We find that it can show a very large diode effect by tuning the crystal axes of $d$-wave superconductors and the magnetization of FI. The difference of the maximum Josephson currents $I_{c}$'s between the positive and negative directions can be about factor 2, where the current-phase relation is modified largely from the conventional one. The relevance of the zero energy Majorana bound states at the interface is also revealed. This result can pave a way to realize an efficient superconducting diode with low energy cost.