Non-Adiabatic Spin Transfer Torque in Realistic Materials

The motion of simple domain walls and more complex magnetic textures in the presence of a transport current is sensitive to the size and sign of the non-adiabatic spin transfer torque coefficient $\beta$, even though this dimensionless coefficient is believed to typically have a small value. The ratio of $\beta$ to the Gilbert damping coefficient $\alpha$ is particularly important and has been variously estimated to be close to 0, close to 1, and large or small. By identifying $\beta$ as following from the influence of a transport current on $\alpha$, we derive concise analytical expressions for $\beta$ in real materials. When spin-orbit is included in the band structure, the damping has an intra-band contribution that is proportional to the square of the quasiparticle lifetime. We will discuss estimates for $\beta$ and for the $\alpha/\beta$ ratio in common magnetic materials.