Determining complex spin mixing conductance and spin diffusion length from spin pumping experiments in magnetic insulator/heavy metal bilayers

Magnetic insulators are promising materials for the development of energy-efficient spintronics. Unlike metallic counterparts, the magnetic insulators are characterized by the imaginary part of the interfacial spin mixing conductance as well in a bilayer with heavy metals, and it is responsible for the field-like toque in spin-orbit torque devices. Here, we study the underlying theoretical constructs and develop a general strategy to determine the complex spin mixing conductance from the experimental results of ferromagnetic resonance and spin pumping. The results show that the imaginary part of the spin mixing conductance can be one order more than the real part and it matches the critical trend of spin mixing conductance with thickness of the heavy metal. The interpretation of experimental results also indicates that at small thicknesses, the interface contribution becomes significant and a bulk diffusion model cannot explain the results. A thickness-dependent spin diffusion length is necessary too that is tantamount to the Elliott-Yafet spin relaxation mechanism in the heavy metals. Also, we effectively explain the experimental results while inserting a copper layer with varying thicknesses in between the magnetic insulator and the heavy metal using spin-circuit formalism.