Orientation-dependent transport in junctions formed by $d$-wave altermagnets and $d$-wave superconductors

We investigate de Gennes-Saint-James states and Josephson effect in hybrid junctions based on $d$-wave altermagnet and $d$-wave superconductor. Even though these states are associated to long junctions, we find that the $d_{x^{2}-y^{2}}$-altermagnet in a normal metal/altermagnet/$d$-wave superconductor junction forms de Gennes-Saint-James states in a short junction due to an enhanced mismatch between electron and hole wave vectors. As a result, the zero-bias conductance peak vanishes and pronounced resonance spikes emerge in the subgap conductance spectra. By contrast, the $d_{xy}$-altermagnet only features de Gennes-Saint-James states in the long junction. Moreover, the well-known features such as V-shape conductance for $d_{x^2-y^2}$ pairings and zero-biased conductance peak for $d_{xy}$ pairings are not affected by the strength of $d_{xy}$-altermagnetism in the short junction. We also study the Josephson current-phase relation $I(\varphi)$ of $d$-wave superconductor/altermagnet/$d$-wave superconductor hybrids, where $\varphi$ is the macroscopic phase difference between two $d$-wave superconductors. In symmetric junctions, we obtain anomalous current phase relation such as a $0$-$\pi$ transition by changing either the orientation or the magnitude of the altermagnetic order parameter and dominant higher Josephson harmonics. Interestingly, we find the first-order Josephson coupling in an asymmetric $d_{x^{2}-y^{2}}$-superconductor/altermagnet/$d_{xy}$-superconductor junction when the symmetry of altermagnetic order parameter is neither $d_{x^{2}-y^{2}}$- nor $d_{xy}$-wave. We present the symmetry analysis and conclude that the anomalous orientation-dependent current-phase relations are ascribed to the peculiar feature of the altermagnetic spin-splitting field.