Motivated by recent advances in utilizing two important classes of two-dimensional materials- topological insulators and transition-metal dichalcogenides (TMDCs)-as spin sources to generate spin-orbit torque (SOT), the applied current-induced spin polarization and the resulting spin-orbit torque in a TMDC and chromium iodide (CrI₃) bilayer, with WSe₂ and MoSe₂ as TMDCs, is investigated beyond the linear response theory.

Making use of the steady-state Boltzmann equation, we find that the spin polarization associated with intra-band transitions generates a strong in-plane, field-like SOT on the ferromagnetic CrI₃ layer, whereas the intrinsic inter-band transitions lead to a weak in-plane, damping-like SOT in WSe₂/CrI₃ bilayers.

Importantly, it is demonstrated that the strength of the damping-like torque can be significantly enhanced by three orders of magnitude in the case of n-doped MoSe₂ and be approximately at the same order as the field-like term, which itself exceeds that of the WSe₂-based bilayer by one order of magnitude. Both the field-like and damping-like SOTs demonstrate large discrepancies between the n-type and p-type doping for WSe₂- and MoSe₂-based bilayers.

Remarkably, depending on the TMDC material and the chemical potential, twisting can lead to a sign change as well as an attenuation or a significant amplification of the strength of the in-plane spin-orbit torque. In addition, we reveal a rather high tunability of the strength of the SOT, about one order of magnitude, as well as a sign change at the twist angle 10.1583° by applying a transverse gate electric field.

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