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Non-Adiabatic Spin Transfer Torque in Real Materials

The motion of simple domain walls and of more complex magnetic textures in the presence of a transport current is described by the Landau-Lifshitz-Slonczewski (LLS) equations. Predictions of the LLS equations depend sensitively on the ratio between the dimensionless material parameter $β$ which characterizes non-adiabatic spin-transfer torques and the Gilbert damping parameter $α$. This ratio has been variously estimated to be close to 0, close to 1, and large compared to 1. By identifying $β$ as the influence of a transport current on $α$, we derive a concise, explicit and relatively simple expression which relates $β$ to the band structure and Bloch state lifetimes of a magnetic metal. Using this expression we demonstrate that intrinsic spin-orbit interactions lead to intra-band contributions to $β$ which are often dominant and can be (i) estimated with some confidence and (ii) interpreted using the "breathing Fermi surface" model.