Topological superconductivity and nonreciprocal edge supercurrent in altermagnetic planar Josephson junctions

Recently, altermagnetism, an unconventional magnetic phase with anisotropic spin-polarization, but zero net magnetization, has garnered significant interest. In this Letter, we report on the topological superconductivity in a d-wave altermagnetic planar Josephson junction (APJJ) featuring strong Rashba spin-orbit coupling, which hosts Majorana zero modes (MZMs) at its ends. By analyzing the Andreev spectra of the Bogoliubov-de-Gennes (BdG) Hamiltonian, we derive the topological phase diagram with respect to the crystallographic orientation angle $\theta$ of the altermagnet and the phase difference Φ between the two superconductors. Notably, robust MZMs appear and disappear within the topological gaps around Φ=π for d_{x²-y^{2 -}}wave (θ=0) and d_xy-wave (θ=π/4) altermagnets, respectively. This behavior is driven by the edge magnetization profile, which protects MZMs in the former case but not in the latter. Consequently, a topological phase transition occurs between them near θ=θ/8. Additionally, a surprising edge Josephson diode effect can emerge due to the edge magnetization and spin-orbit coupling, while the C₄ symmetry of d-wave altermagnets prohibits nonreciprocal supercurrents in the bulk of the APJJ. Through utilizing the interplay between altermagnetism and superconductivity, our discovery provides new opportunities for field-free topological quantum computing and superconducting electronics.