Spin transfer torque with spin diffusion in magnetic tunnel junctions

Spin transport in magnetic tunnel junctions in the presence of spin diffusion is considered theoretically. Combining ballistic tunneling across the barrier and diffusive transport in the electrodes, we solve the spin dynamics equation in the metallic layers. We show that spin diffusion mixes the transverse spin current components and dramatically modifies the bias dependence of the effective spin transfer torque. This leads to a significant linear bias dependence of the out-of-plane torque, as well as a nonconventional thickness dependence of both spin torque components.

Figure 1

Transverse spin accumulation as a function of the distance in the electrode in $F$(2 nm)/$N$. The parameters are ${\lambda }_J$ = 1 nm, ${\lambda }_{sf}^F$ = 15 nm, ${\lambda }_{sf}^N$ = 2 nm, and ${\mathcal{D}}_F/{\mathcal{D}}_N=1$.

Figure 2

Thickness dependence of the normalized spin torques $d{T}_{\parallel }$ and $d{T}_{\perp }$ for different spin precession lengths, ${\lambda }_J=0.5$ nm, 1 nm, and 1.5 nm. The parameters are the same as described in the legend of Fig. 1, except for the spin dephasing length: ${\lambda }_{\phi }=0.5$ nm (solid lines) and ${\lambda }_{\phi }$ = 1 nm (dotted lines). Inset: thickness dependence of the ratio between out-of-plane torque and in-plane torque for the same parameters.

Figure 3

Bias dependence of the in-plane (a) and out-of-plane (b) torques for different thicknesses of the ferromagnetic layer. The parameters are ${\lambda }_J={\lambda }_{\phi }=$1 nm, ${\lambda }_{sf}^F$ = 10 nm, ${\lambda }_{sf}^N$ = 1 nm, and ${\mathcal{D}}_F/{\mathcal{D}}_N$ = 1. The bias dependence of the interfacial spin current ${\mathcal{J}}_0$ is taken from the spin torque bias dependence measured by Sankey et al.[14] The inset display the zoom in the range $\pm 200$ mV.