Motivated by the orientation-dependent properties of $d$-wave superconductors (SCs), we investigate Andreev bound states (ABSs) and Josephson current in $s$-wave SC/altermagnet/$d$-wave SC (S/AM/D) and $d$-wave SC/altermagnet/$d$-wave SC (D/AM/D) junctions. The asymmetric S/AM/D junction exhibits a nodeless ABS spectrum with distinct spin states, arising from AM manipulation. In contrast, a symmetric D/AM/D junction with $\pm {45}^{\circ }$-oriented order parameters exhibits a nodal ABS spectrum, characterized by spin-split sinusoidal curves. This behavior arises from the angular dependence of the $d$-wave pairing potential, which introduces a $\pi$ shift in the ABS spectrum of the D/AM/D junction compared to that of conventional S/AM/S junctions. Additionally, the positions of the ABS nodes in the D/AM/D junction can be controlled by adjusting the length and strength of the AM layer. Analysis of the free energy and Josephson current reveals additional extrema in the free energy at intermediate phase differences, leading to skewness and a nonsinusoidal behavior in the current-phase relationship of both configurations. These zero-supercurrent phase differences can be tuned by adjusting the properties of the AM layer such as AM strength and length. These findings provide insights into AM-based Josephson junctions.

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• Andreev bound states
• Andreev reflection
• Josephson effect
• Superconductivity
• Altermagnets
• Heterostructures
• Superconductors