Magnetic and Crystal Symmetry Control on Spin Hall Conductivity in Altermagnets

Altermagnets combine zero net magnetization with spin splitting, opening new opportunities for next-generation spintronic devices. In this work, the unconventional spin Hall conductivity (USHC) in three representative materials-RuO₂, CrSb, and MnTe is explored. It is clarified how distinct magnetic and crystal symmetries modulate their spin Hall responses. RuO₂ exhibits only trivial USHC under a tilted geometry, demonstrating that symmetry projections alone can induce apparent unconventional elements. In contrast, CrSb and MnTe manifest robust intrinsic USHC driven by symmetry reduction through easy-axis magnetic ordering without structural tilts. Through extensive first-principles calculations, the complementary roles of the time-reversal ( $T$ )-even and $T$ -odd components in determining the overall spin Hall conductivity are demonstrated. The findings indicate that the interplay between crystal and magnetic symmetry can be controlled through structural tilting and magnetic axis orientation. These results pave the way for the engineering of multifunctional spintronic devices. In particular, they highlight zero-net-moment materials with tunable spin configurations as promising platforms for coherent and robust spin transport.