Emergence of g -Wave Altermagnetism in Three-Dimensional Metal–Organic Frameworks

Altermagnets integrate key characteristics of both antiferromagnets and ferromagnets, exhibiting a vanishing net magnetic moment while breaking time-reversal symmetry and displaying momentum-dependent spin splitting in the absence of an external field. Recognized as one of the top ten Science breakthroughs of 2024, altermagnetism has drawn increasing attention, particularly in inorganic systems. In this work, using density functional theory calculations, we identify a g-wave altermagnetic state in thermally stable three-dimensional (3D) metal-organic frameworks (MOFs) based on Co(pymo)₂ and its halogen-substituted derivatives, thereby extending the scope of altermagnetism to organic-containing 3D frameworks. A range of magnetic configurations was systematically evaluated for these systems, with the altermagnetic state being consistently found as the most energetically favorable. This feature is consistent with the antiferromagnetic coupling experimentally observed between Co ions connected via ligand bridges. The momentum-dependent spin splitting in the electronic structure emerges independently of spin-orbit coupling and is not significantly enhanced through the inclusion of heavy halogen elements. Additional canted spin configurations were also explored, which confirmed the robustness of the spin splitting against fluctuations. These results not only establish a momentous presence of g-wave altermagnetism in these 3D MOFs but also highlight the potential of molecular framework design for realizing symmetry-driven spin phenomena in a class of 3D materials beyond conventional inorganic systems.