Gradient Expansion Formalism for Generic Spin Torques*

Spin torques have been studied in the field of spintronics since the celebrated discovery of the current-induced magnetization reversal by the spin transfer torque (STT). In addition to the STT, the so-called β-term arises from spin relaxation. In the presence of spin-orbit interactions (SOIs), the spin-orbit torque arises even without magnetic textures. In real materials, both magnetic textures and SOIs exist, and hence a systematic formalism for spin torques is desired.
We propose a new quantum-mechanical formalism for spin torques based on the gradient expansion [1], which naturally involves spacetime gradients of the magnetization and electromagnetic fields. We have no assumption in the small-amplitude formalism [2] or no difficulty in the SU(2) gauge transformation formalism [3]. As a representative, we calculate the spin renormalization, Gilbert damping, STT, and β-term in a three-dimensional ferromagnetic metal by taking into account nonmagnetic and magnetic impurities. Our results serve as a first-principles formalism for spin torques.

[1] A. Shitade, arXiv:1708.03424.
[2] H. Kohno et al., J. Phys. Soc. Jpn. 75, 113706 (2006).
[3] H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710 (2007).