Dynamic magnetic susceptibility and electrical detection of ferromagnetic resonance

The dynamic magnetic susceptibility of magnetic materials near ferromagnetic resonance (FMR) is very important in interpreting the dc voltage obtained in its electrical detection. Based on the causality principle and the assumption that the usual microwave absorption lineshape of a homogeneous magnetic material around FMR is Lorentzian, the general forms of the dynamic magnetic susceptibility of an arbitrary sample and the corresponding dc voltage lineshapes of its electrical detection were obtained. Our main findings are as follows. (1) The dynamic magnetic susceptibility is not a Polder tensor for a material with an arbitrary magnetic anisotropy. The two off-diagonal matrix elements of the tensor near FMR are not, in general, opposite to each other. However, the linear response coefficient of the magnetization to the total radio frequency (rf) field (the sum of the external and internal rf fields due to precessing magnetization is a quantity which cannot be measured directly) is a Polder tensor. This may explain why the two off-diagonal susceptibility matrix elements were always wrongly assumed to be opposite to each other in almost all analyses. (2) The frequency dependence of dynamic magnetic susceptibility near FMR is fully characterized by six real numbers, while its field dependence is fully characterized by seven real numbers. (3) A recipe of how to determine these numbers by standard microwave absorption measurements for a sample with an arbitrary magnetic anisotropy is proposed. Our results allow one to unambiguously separate the contribution of the anisotropic magnetoresistance to the dc voltage signals from the anomalous Hall effect. With these results, one can reliably extract the information of spin pumping and the inverse spin-Hall effect, and determine the spin-Hall angle. (4) In the case that resonance frequency is not sensitive to the applied static magnetic field, the field dependence of the matrix elements of dynamic magnetic susceptibility, as well as the dc voltage, may have another non-resonance broad peak. Thus, one should be careful in interpreting the observed peaks.