Anatomy of Spin--Orbit Torques in Monolayer Fe$_3$GeTe$_2$ and Fe$_3$GaTe$_2$: Insights from atomistic and momentum-space decompositions

Abstract

We present a systematic first-principles study of the spin-orbit torques in the ferromagnetic monolayers Fe$_3$GeTe$_2$ (FGT) and Fe$_3$GaTe$_2$ (FGaT). Despite sharing the same crystal structure (point group $D_{3h}$) and predominantly Fe~$3d$ spin-polarized bands, the two materials exhibit markedly different current-induced torques. We reveal these differences by computing the full angular dependence of the torkance---the torque per unit applied electric field---using linear-response theory with symmetry-adapted spin--orbit-coupled Wannier functions. FGaT may be viewed as a hole-doped analogue of FGT, since Ga contributes one valence electron fewer than Ge. Although the work functions differ by only about $28$~meV, the band filling near $K$ and $K'$ changes substantially: the density of states at $\varepsilon_F$ is reduced by a factor of three and its spin polarization reverses from majority in FGT to minority in FGaT. These electronic changes are reflected in the torques resolved by time-reversal parity, sublattice, and momentum. In particular, we identify pronounced hidden torques in FGaT and relate the suppression of its fourth-harmonic Fermi-sea component to the evolution of momentum-space pockets. Finally, we discuss the emergence of such self-torques, which are not captured by the conventional picture of current-induced spin accumulation, within a symmetry-based phenomenological framework. Our results provide microscopic insight into current-induced torques in two-dimensional ferromagnets and offer guidance for defect and van der Waals engineering of layered magnetic materials.