Strong higher-order angular dependence of spin-orbit torque in W/CoFeB bilayer

The symmetry of normal metal/ferromagnet bilayers allows spin-orbit torques (SOTs) to simultaneously have two distinct angular dependences on the magnetization direction $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{m}}$. The most well-studied forms of SOT consist of the conventional fieldlike and dampinglike torques, which we label as ``lowest-order'' SOT. There are additional SOT forms associated with spin polarization different from that of the lowest-order SOT, and which contain an extra factor of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{m}}$ dependence. We label these as ``higher-order'' SOT. Understanding SOT-driven magnetization dynamics requires detailed information about the full angular dependence. In this paper, we measure both the lowest-order and higher-order angular dependences of SOTs in three types of bilayers, Pt/Co, Ta/CoFeB, and W/CoFeB, using harmonic Hall measurements. It is found that the higher-order SOT is negligible for Pt/Co and Ta/CoFeB, whereas it is dominant over the lowest-order one for W/CoFeB. Macrospin simulations show that the higher-order SOT can significantly affect the magnetization dynamics, which is qualitatively in line with SOT-induced switching experiments.