Microwave-assisted domain-wall motion induced by alternating spin-polarized current

We have studied the microwave-assisted domain-wall (DW) motion driven by the alternating spin-polarized current. It is found that the adiabatic spin-transfer torque alone can sustain a steady-state wall motion without Walker breakdown if a transverse microwave field is applied. The direction and velocity of the domain-wall motion can be effectively manipulated by tuning the phase difference between the ac and the microwave field. The wall motion is very sensitive to the microwave frequency and can be resonantly enhanced by exciting the DW thickness oscillation mode. The characteristics of this kind of DW motion are well accounted for by an analytical model.

Figure 1

Illustration of a 180° Bloch DW structure positioned at the position $y$ = 0. The arrows represent the magnetization direction. The Cartesian coordinate system is shown on the higher left, the polar angle is θ (the polar axis being the $z$ axis), and the azimuthal angle φ describes the orientation of the magnetization projection on the $x$-$y$ plane.

Figure 2

(a) The wall displacement $q$ as a function of time $t$ induced by an ac for β = 0 and $u_0$ = 50 m/s with and without the assistance of the microwave field. The simulated results are curve 1 ($h_0$ = 0 mT), curve 2 ($h_0$ = 0.5 mT, $f$ = 6 GHz, and ${\rho }_1$ = π/2), curve 3 ($h_0$ = 0.5 mT, $f$ = 6 GHz, and ${\rho }_1$ = −π/2), and curve 4 ($h_0$ = 0.5 mT, $f$ = 5.5 GHz, and ${\rho }_1$ = π/2). The theoretically calculated result is depicted by curve 5 ($h_0$ = 0.5 mT, $f$ = 6 GHz, and ${\rho }_1$ = π/2) and curve 6 ($h_0$ = 0.5 mT, $f$ = 6 GHz, and ${\rho }_1$ = −π/2). (b) Simulated DW displacement for varying β [curve 1 (β = 0), curve 2 (β = α/2), curve 3 (β = α), and curve 4 (β = 2α)] when $u_0$ = 50 m/s, $h_0$ = 0.5 mT, $f$ = 6 GHz, and ${\rho }_1$ = π/2. The analytically calculated DW displacement for β = 2α is curve 5. The inset of (b) shows the simulated (black squares) and analytically calculated (blue circles) steady-state wall motion velocity $v_s$ as a function of β − α.

Figure 3

(a) Simulated (black squares for β = 0 and red regular triangle for β = 2α) and analytically calculated (blue circles for β = 0 and magenta inverse triangle for β = 2α) steady-state wall motion velocity $v_s$ as a function of the ac phase ${\rho }_1$ when $f$ = 6 GHz, $h_0$ = 0.5 mT, and $u_0$ = 50 m/s. (b) Simulated (black squares) and analytically calculated (blue circles) steady-state wall motion velocity $v_s$ as a function of the frequency $f$ when $h_0$ = 0.5 mT, $u_0$ = 50 m/s, β = 0, and ${\rho }_1$ = π/2. Resonance frequency spectrum of the DW excited by the transverse microwave field is also shown in (b).

Figure 4

(a) Simulated (black squares) and analytically calculated (blue circles) steady-state wall motion velocity $v_s$ as a function of the ac amplitude $u_0$ when $f$ = 6 GHz, $h_0$ = 0.5 mT, ${\rho }_1$ = π/2, and β = 0. (b) Simulated (black squares) and analytically calculated (blue circles) steady-state wall motion velocity $v_s$ as a function of the microwave field amplitude $h_0$ when $f$ = 6 GHz, $u_0$ = 50 m/s, ${\rho }_1$ = π/2, and β = 0. The inset of (b) shows simulated DW thickness oscillation amplitude Δ₁ (black triangles) and ${\phi }_{20}$ (red circles) as a function of $h_0$.

Figure 5

The simulated wall displacement $q$ in the realistic Py nanostrip as a function of time $t$ induced by an ac for $u_0$ = 50 m/s and β = 0 with and without the assistance of the microwave field. Curve 1 is the displacement without the microwave field. Curves 2, 3, and 4 represent the displacement assisted by the microwave field with phase difference ${\rho }_1$ = π/2, 3π/4, and −π/2, respectively. The inset shows the simulated (β − α) dependence of the wall motion velocity $v_s$ for the realistic Py nanostrip.