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

In the normal-metal–ferromagnetic-insulator bilayer (such as $\mathrm{Pt}/{\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$) and the normal-metal–ferromagnetic-metal–oxide trilayer (such as $\mathrm{Pt}/\mathrm{Co}/{\mathrm{AlO}}_x$) 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 dampinglike to fieldlike component depends on the tunneling wave function that is strongly influenced by generic material properties such as interface $s\text{−}d$ 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.

Figure 1

(a) Definition of the coordinates at $\phi =0$ and energy scales near the NM-FMI interface. Blue line indicates the potential profile seen by the conduction electrons. (b) The ratio of (c) the dampinglike to (d) the fieldlike component of the spin mixing conductance at $\theta =0.3\pi$ plotted as a function of the insulating gap $(V_0-\epsilon )/\epsilon$ and interface $s\text{−}d$ coupling $-\mathrm{\Gamma }S/\epsilon$ relative to the Fermi energy. The color scale is in units of $e^2/\hslash a^2$ which is about $10^{-15}-10^{-14}{\mathrm{\Omega }}^{-1}{\mathrm{m}}^{-2}$ depending on the lattice constant $a$. The white regions in (b)–(d) are where one potential barrier is lower than the Fermi energy $V_0-\epsilon +\mathrm{\Gamma }S<\epsilon$, hence, irrelevant to the problem. If the effective mass in NM and in FMI are different, the axis labels of the plots are replaced by $-\mathrm{\Gamma }S/\epsilon \to -m_F\mathrm{\Gamma }S/m_N\epsilon$ and $(V_0-\epsilon )/\epsilon \to m_F(V_0-\epsilon )/m_N\epsilon$.

Figure 2

Same as Fig. 1, but for the NM-FMM-oxide trilayer. (b) to (d) are plotted in terms of the $s\text{−}d$ coupling relative to the Fermi energy $-\mathrm{\Gamma }S/\epsilon$ and the thickness of the FMM layer $b/a$.