Existence of vertical spin stiffness in Landau-Lifshitz-Gilbert equation in ferromagnetic semiconductors

TL;DR

This paper demonstrates the existence of a vertical spin stiffness term in the Landau-Lifshitz-Gilbert equation for ferromagnetic semiconductors, which influences domain-wall structures and magnetization dynamics.

Contribution

It derives a new vertical spin stiffness term from kinetic spin Bloch equations, highlighting its importance in modeling ferromagnetic semiconductor dynamics.

Findings

Vertical spin stiffness term modifies domain-wall structure.

Effective magnetic field perpendicular to spin stiffness is proportional to β.

Vertical spin stiffness should be included in LLG equation for accurate modeling.

Abstract

We calculate the magnetization torque due to the spin polarization of the itinerant electrons by deriving the kinetic spin Bloch equations based on the $s$-$d$ model. We find that the first-order gradient of the magnetization inhomogeneity gives rise to the current-induced torques, which are consistent to the previous works. At the second-order gradient, we find an effective magnetic field perpendicular to the spin stiffness filed. This field is proportional to the nonadiabatic parameter $\beta$. We show that this vertical spin stiffness term can significantly modify the domain-wall structure in ferromagnetic semiconductors and hence should be included in the Landau-Lifshitz-Gilbert equation in studying the magnetization dynamics.