Altermagnets possess two key features: nonrelativistic alternating spin splitting (i.e., altermagnetism) and a material-dependent Néel vector. The former naturally coexists with Rashba spin-orbit coupling (SOC) in real materials on substrates, prompting the question of how SOC affects the magnetic properties of altermagnets. The latter is crucial for information storage, making it essential to determine its orientation. To address these issues, we study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in two-dimensional (2D) $d$-wave altermagnets by independently varying the Néel vector orientation and the SOC strength. Our results demonstrate that the Néel vector orientation can be accurately determined via the Ising term without SOC or qualitatively inferred via the Dzyaloshinskii-Moriya (DM) terms with SOC. Moreover, we observe a DM component distinct from previous reports, whose emergence is attributed to the synergy between altermagnetism and SOC. Additionally, by tuning the SOC strength, we reveal the evolution of the RKKY spin models governed by five distinct mechanisms: The spin model may be determined solely by altermagnetism, solely by SOC, or solely by the kinetic term; alternatively, altermagnetism may coincidentally yield the same moderately anisotropic spin model as SOC or compete with SOC to produce a spin model with maximal anisotropy. Beyond SOC strength, which mechanism operates also relies on the Néel vector orientation and impurity configurations. All results are numerically verified. These findings—which were inaccessible in prior studies due to the limitations of first-order SOC expansion and fixed Néel vector orientation—provide important insights into the magnetic properties of altermagnets.

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• RKKY interaction
• Spin-phonon coupling
• Altermagnets
• Green's function methods
• Perturbation theory