Interplay of altermagnetism and weak ferromagnetism in two-dimensional RuF$_4$

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 RuF$_4$ monolayer is a two-dimensional $d$-wave altermagnet. Spin-orbit coupling leads to pronounced spin splitting of the electronic bands at the $\Gamma$ point by $\sim 100$ meV and turns the RuF$_4$ into a weak ferromagnet due to non trivial 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 non trivial spin-momentum locking and weak ferromagnetism in the two-dimensional altermagnets relevant for novel venues in this emerging field of material science research.