A Drift-Diffusion Based Modeling and Optimization Framework for Nanoscale Spin-Orbit Torque Devices

We present a comprehensive set of experimentally validated/calibrated models that capture the physics of the nanoscale spin-orbit torque (SOT) devices. We consider various effects that are prominent at nanoscale including incomplete current redistribution, interface spin mixing, and nonuniform resistivity that were ignored in the prior modeling efforts. We develop a formalism based on drift-diffusion equations and the transfer matrix method to accurately estimate spin current distribution. We utilize finite element simulations to accurately calculate electric current density and field, and see considerable differences to the results from the simplified lumped model commonly used. For example, we calculate ~20% smaller spin current in a 15-nm-wide ferromagnet with a 4-nm-thick
$\beta $
-W SOT channel. We further account for the effect of interface scattering on resistivity. Finally, we quantify the optimal SOT-layer thickness that minimizes the write energy as a function of spin diffusion length and conductivity of SOT materials. We show that the optimal thickness increases with the spin diffusion length and decreases with the conductivity.