We study phase-controlled planar Josephson junctions comprising a two-dimensional electron gas with strong spin-orbit coupling and $d$-wave superconductors, which have an advantage of a high critical temperature. We show that a region between the two superconductors can be tuned into a topological state by the in-plane Zeeman field, and can host Majorana bound states. The phase diagram as a function of the Zeeman field, chemical potential, and the phase difference between superconductors exhibits the appearance of Majorana bound states for a wide range of parameters. We further investigate the behavior of the topological gap and its dependence on the type of $d$-wave pairing, i.e., $d, d+is$, or $d+i{d}^{\prime }$, and note the difficulties that can arise due to the presence of gapless excitations in pure $d$-wave superconductors. On the other hand, the planar Josephson junctions based on superconductors with $d+is$ and $d+i{d}^{\prime }$ pairings can potentially lead to realizations of Majorana bound states. Our proposal can be realized in cuprate superconductors, e.g., in a twisted bilayer, combined with the layered semiconductor ${\mathrm{Bi}}_2{\mathrm{O}}_2\mathrm{Se}$.

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• Majorana bound states
• Topological superconductors
• d-wave
• Josephson junctions