Unconventional magnetoresistance and its implications in spin-torque characterization*

Magnetoresistance (MR) is essential in characterizing spin torques, especially in spin-torque ferromagnetic resonance (ST-FMR) and harmonic Hall analysis. Some experiments utilize a ferromagnetic (FM) metal that hosts anisotropic MR, where current in an adjacent material generates spin torques exerting on the magnetization and changes the MR. The variation of MR can in turn be used to quantify the spin torques, even when the spin-torque generation is attributed to unconventional mechanisms such as the out-of-plane anti-damping torque recently discovered in WTe₂. A different type of experimental setup involves an insulating FM driven and detected by a heavy metal. In such a system, there is no shunting current in the FM insulator, so the spin Hall effect in the heavy metal generates the spin torque and at the same time detects the FM dynamics through the change of the spin Hall MR. However, when it comes to an FM insulator driven by a non-spin-Hall system, such as WTe₂/Cr₂Ge₂Te₆ bilayer, there has been no clear understanding of the MR, let alone using the MR to characterize the spin torques. In this work, we first investigate the unconventional form of MR in a non-spin-Hall geometry arising from the spin diffusion effect in the presence of spin backflow. Next, by solving the Landau-Lifshitz-Gilbert equation in combination with the harmonic expansion of the unconventional MR, we derive a series of expressions for measurable ST-FMR and harmonic Hall signals, which are compared with conventional theories based on the spin Hall MR. Verified by detailed numerical results, our finding lays the theoretical foundation for characterizing spin torques originating from unconventional mechanisms beyond the spin Hall effect, which can not only guide ongoing experimental endeavors but also stimulate future experimental designs.