Anatomy of electrical signals and dc-voltage line shape in spin-torque ferromagnetic resonance

The electrical detection of spin-torque ferromagnetic resonance (st-FMR) is becoming a popular method for measuring the spin-Hall angle of heavy metals (HM). However, various sensible analysis on the same material with either the same or different experimental setups yielded different spin-Hall angles with large discrepancy, indicating some missing ingredients in our current understanding of st-FMR. Here we carry out a careful analysis of electrical signals of the st-FMR in a HM/ferromagnet (HM/FM) bilayer with an arbitrary magnetic anisotropy. The FM magnetization is driven by two radio-frequency (rf) forces: the rf Oersted field generated by an applied rf electric current and the so called rf spin-orbit torque from the spin current flowing perpendicularly from the HM to the FM due to the spin-Hall effect. By using the universal form of the dynamic susceptibility matrix of magnetic materials at the st-FMR, the electrical signals, originated from the anisotropic magnetoresistance, anomalous Hall effect, and inverse spin-Hall effect are analyzed and dc-voltage line shapes near the st-FMR are obtained. Angular dependence of dc voltage is given for two setups. A way of experimentally extracting the spin-Hall angle of a HM is proposed.