Circuit model for domain walls in ferromagnetic nanowires:  Application to conductance and spin transfer torques

We present a circuit model to describe the electron transport through a domain wall in a ferromagnetic nanowire. The domain wall is treated as a coherent four-terminal device with incoming and outgoing channels of spin up and down and the spin-dependent scattering in the vicinity of the wall is modeled using classical resistances. We derive the conductance of the circuit in terms of general conductance parameters for a domain wall. We then calculate these conductance parameters for the case of ballistic transport through the domain wall, and obtain a simple formula for the domain wall magnetoresistance, which gives a result consistent with recent experiments. The spin transfer torque exerted on a domain wall by a spin-polarized current is calculated using the circuit model, and an estimate of the speed of the resulting wall motion is made.

Figure 1

The two-resistor model applied to an interface between two ferromagnetic layers, in which the magnetization is (a) parallel and (b) antiparallel.

Figure 2

The circuit model for a domain wall used in this paper.

Figure 3

Transmission functions $T_{\uparrow \uparrow }\left(ϵ\right)$ (solid lines) and $T_{\uparrow \downarrow }\left(ϵ\right)$ (dashed lines) as a function of $ϵ(=2E_z∕\Delta )$ for $p$ equal to (a) 2.5, (b) 5, and (c) 20.

Figure 4

Adiabaticity parameter $P$ (solid lines) and $1-P$ (dashed lines) as a function of the domain wall width parameter $p$ for ${ϵ}_F=10$ (thin lines) and ${ϵ}_F=100$ (thick lines).

Figure 5

The magnetoresistance, as a fraction of the GMR interface value $(P=0)$, for $\alpha =1$, 10, and 80.