and confers an out of plane spin accumulation to the charge carriers

Physically, the origin of the $(\hat{b}\times(E_x \partial_{k_x}\hat{b})$ term can be explained by assuming that the spins of the charge carriers adiabatically follow the direction of the SOI field. As shown in Fig. \[gA2SOIfield\], $\vec{B}(\vec{k})\cdot\vec{\sigma}$ associates each point in $k$ space with a SOI field pointing in the $\hat{b}(\vec{k})$ direction. Assume that the electric field is initially switched off and consider a carrier with a definite $\vec{k}$. As the electric field is switched on, the field causes the charge carrier to accelerate in the direction of the field so that the momentum changes and the carrier traces out a trajectory along $k$ space. We assume that the electric field is weak enough so that the spin of the carrier rotates along with the direction of the SOI field as it successively moves through different $k$ points. The resulting rotation of the spin can be thought of as being due to an effective magnetic field pointing along the $\hat{b}\times\partial_t \hat{b} =  E_x \hat{b}\times\partial_{k_x} \hat{b}$ direction which both provides the torque necessary to rotate the spin as well as confers a spin accumulation in the direction of the torque.

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