Inspired by recent advances in utilizing topological insulators and transition-metal dichalcogenides (TMDCs) as spin sources for generating spin-orbit torque (SOT), we explore the current-induced spin polarization and resulting SOT in a TMDC/CrI₃ bilayer-specifically with WSe₂ and MoSe₂â??beyond the linear response theory.

Using the steady-state Boltzmann equation, we find that non-linear spin polarization associated with intra-band transitions generates a strong in-plane SOT on the ferromagnetic CrI₃ layer, whereas the contribution of intrinsic inter-band and intra-band transitions is zero in the linear response regime.

The SOT demonstrates an odd function with respect to the magnetization-called the field-like torque-a large discrepancy between n-type and p-type doping for both TMDC/CrI₃ bilayers, and sign changes in MoSe₂- and p-type doped WSe₂-based structures. Remarkably, the SOT strength in the n-doped MoSe₂/CrI₃ bilayer is an order of magnitude larger than in the WSe₂/CrI₃ case.

We further show that, depending on the TMDC material and chemical potential, twisting can cause the appearance or disappearance of additional sign changes, as well as attenuation or significant amplification of the in-plane SOT. Specifically, the WSe₂-based bilayer exhibits high SOT for p-type doping at lower chemical potentials, while the MoSe₂-based structure shows large SOT for n-type doping at higher chemical potentials.

In addition, we reveal a high tunability of the SOT strength-about one order of magnitude-by applying a transverse gate electric field.