Minimal Model of Spin-Transfer Torque and Spin Pumping caused by Spin Hall Effect

In the normal metal/ferromagnetic insulator bilayer (such as Pt/Y$_{3}$Fe$_{5}$O$_{12}$) and the normal metal/ferromagnetic metal/oxide trilayer (such as Pt/Co/AlO$_{x}$) where spin injection and ejection are achieved by the spin Hall effect in the normal metal, we propose a minimal model based on quantum tunneling of spins to explain the spin-transfer torque and spin pumping caused by the spin Hall effect. The ratio of their damping-like to field-like component depends on the tunneling wave function that is strongly influenced by generic material properties such as interface $s-d$ coupling, insulating gap, and layer thickness, yet the spin relaxation plays a minor role. The quantified result renders our minimal model an inexpensive tool for searching for appropriate materials.