Interplay of altermagnetism and weak ferromagnetism in two-dimensional RuF4

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Gaining growing attention in spintronics is a class of magnets displaying zero net magnetization and spin-split electronic bands called altermagnets. Here, by combining density functional theory and symmetry analysis, we show that RuF4 monolayer is a two-dimensional (2D) d-wave altermagnet. Spin–orbit coupling leads to pronounced spin splitting of the electronic bands at the Γ point by ∼100meV and turns the RuF4 into a weak ferromagnet due to nontrivial spin-momentum locking that cants the Ru magnetic moments. The net magnetic moment scales linearly with the spin–orbit coupling strength. Using group theory we derive an effective spin Hamiltonian capturing the spin-splitting and spin-momentum locking of the electronic bands. Disentanglement of the altermagnetic and spin–orbit coupling induced spin splitting uncovers to which extent the altermagnetic properties are affected by the spin–orbit coupling. Our results move the spotlight to the nontrivial spin-momentum locking and weak ferromagnetism in the 2D altermagnets relevant for novel venues in this emerging field of material science research.