Spin-orbit-coupling induced torque in ballistic domain walls: equivalence of charge-pumping and nonequilibrium magnetization formalisms

TL;DR

This paper investigates how spin-orbit coupling affects spin-transfer torque in ballistic magnetic domain walls, demonstrating the equivalence of charge-pumping and nonequilibrium magnetization formalisms and revealing different torque behaviors depending on domain wall length.

Contribution

It introduces and compares two SOC-inclusive formalisms for calculating spin-transfer torque, showing their quantitative agreement and analyzing the length-dependent torque effects.

Findings

Out-of-plane torque increases rapidly for short domain walls due to nonadiabatic reflection.

Long domain walls exhibit a linear increase in the out-of-plane torque parameter with length.

The two formalisms are quantitatively consistent, confirming their equivalence.

Abstract

To study the effect of spin-orbit coupling (SOC) on spin-transfer torque in magnetic materials, we have implemented two theoretical formalisms that can accommodate SOC. Using the "charge-pumping" formalism, we find two contributions to the out-of-plane spin-transfer torque parameter $\beta$ in ballistic Ni domain walls (DWs). For short DWs, the nonadiabatic reflection of conduction electrons caused by the rapid spatial variation of the exchange potential results in an out-of-plane torque that increases rapidly with decreasing DW length. For long DWs, the Fermi level conduction channel anisotropy that gives rise to an intrinsic DW resistance in the presence of SOC leads to a linear dependence of $\beta$ on the DW length. To understand this counterintuitive divergence of $\beta$ in the long DW limit, we use the "nonequilibrium magnetization" formalism to examine the spatially resolved spin-transfer torque. The SOC-induced out-of-plane torque in ballistic DWs is found to be quantitatively consistent with the values obtained using the charge-pumping calculations indicating the equivalence of the two theoretical methods.