Current induced pressure on a tilted magnetic domain wall

We consider the effect of an electrical current on a magnetic domain wall (DW) in conducting ferromagnets. When the current lines are at normal incidence onto the wall, two relevant physical effects, based on the Hall voltage and direct spin transfer, induce a pressure which is able to move DWs. When the current is not at normal incidence, we show that the Hall effect can generate a local charge on the wall which is subjected to a force from the electric field in the sample. The resulting pressure is quadratic in current, thus independent of its sign, and can dominate at high current densities and in materials with large resistivity and Hall angle. Hence, the effect is particularly relevant for magnetic semiconductors.

Figure 1

(Color online) Schematics of the sample geometry where a DW separates two domains of opposite magnetization. The “out-of-plane direction” can be either vertical or horizontal in the figure, i.e. the system is with perpendicular anisotropy and a DW tilted horizontally or $M$ is planar and the DW is tilted within the thickness.

Figure 2

(Color online) Top view of the Hall-induced current and electric field around the tilted DW of Fig. 1 with perpendicular magnetization. The arrows represent the extra contribution when the Hall effect is turned on for: (a) The current lines, which loop around the DW thus generating the perpendicular field responsible for the “Hall drag” effect, and (b) the electric field whose discontinuity at the wall produces the local charges responsible for the “Hall charging” effect.

Figure 3

(Color online) Current density dependence of the pressure generated by the different mechanisms in $50\mathrm{nm}$ thick stripes of Fe, GaMnAs, and InMnAs. The spin torque effect (black line) is roughly the same for all three materials but the domain drag (short lines of open symbols) and Hall charging (longer lines of filled symbols) can vary greatly depending on Hall angle, resistivity, and magnetization values. The hatched regions represent current densities at which DWs move for NiFe $(\text{higher}+\text{red})$ and GaMnAs $(\text{lower}+\text{green})$.

Figure 4

(Color online) Combined effect of the spin torque and Hall pressure in an ideal system: (a) The wall without current is exactly perpendicular to the stripe direction. (b) At low current density, the spin torque pushes the DW, but pinning at the edges prevents motion. (c) At higher current, the Hall pressure charges the wall depending on the local angle between the DW and the current lines. (d) The wall is distorted and tilts which enhances the Hall pressure in a direction opposite to the electrons’ flow.