Width Dependence of the Nonadiabatic Spin-Transfer Torque in Narrow Domain Walls

We analytically determine the spatially varying spin-transfer torque within a domain wall. In the case of ballistic spin and diffusive charge transport, the spin-transfer torque as well as the local degree of nonadiabaticity oscillate within a domain wall. In narrow domain walls, the degree of nonadiabaticity ceases to be a constant material parameter but depends on the domain-wall width including a possible sign change, which is crucial for experiments and the technological utilization in spin-transfer-torque-based storage devices.

Figure 1

Cartesian components of the magnetization of the conduction electrons in units ${\mu }_B{\tau }_{sd}Pj/(e\lambda )$ within domain walls of different width $\lambda$ for $l_{\mathrm{mfp}}=5.5\mathrm{nm}$. (a),(c) $\lambda =200\mathrm{nm}$ and (b),(d) $\lambda =20\mathrm{nm}$. The dashed black line represents the average magnetization, while the blue dotted line depicts its spatially independent part. The frequency of the oscillations is determined by the $sd$ interaction.

Figure 2

Integrated torques versus domain-wall width $\lambda$: (a) Adiabatic and (b) nonadiabatic spin-transfer torque. (c) Nonadiabaticity $\xi$ due to the magnetization texture. The inset depicts a blowup of the regime where $\xi$ changes sign. The solid red line indicates the regime where the perturbation theory starts to be unreliable.