Domain-wall dynamics driven by adiabatic spin-transfer torques

In a first approximation, known as the adiabatic process, the direction of the spin polarization of currents is parallel to the local magnetization vector in a domain wall. Thus the spatial variation of the direction of the spin current inside the domain wall results in an adiabatic spin-transfer torque on the magnetization. We show that domain-wall motion driven by this spin torque has many unique features that do not exist in the conventional wall motion driven by a magnetic field. By analytically and numerically solving the Landau-Lifshitz-Gilbert equation along with the adiabatic spin torque in magnetic nanowires, we find that the domain wall has its maximum velocity at the initial application of the current but the velocity decreases to zero as the domain wall begins to deform during its motion. We have computed domain-wall displacement and domain-wall deformation of nanowires, and concluded that the spin torque based on the adiabatic propagation of the spin current in the domain wall is unable to maintain wall movement. We also introduce the concept of domain-wall inductance to characterize the capacity of the spin-torque-induced magnetic energy stored in a domain wall. In the presence of domain-wall pinning centers, we construct a phase diagram for the domain-wall depinning by the combined action of the magnetic field and the spin current.

Figure 1

Cartesian and polar coordinate systems.

Figure 2

The velocity $v$, the wall width $W$, the wall angle $\phi$, and the displacement $x$ as a function of time $t$ for different spin torques and for $H_{ext}=0\mathrm{Oe}$. The parameters are $4\pi M_s=1.8\times {10}^4\mathrm{Oe}$, $H_K=500\mathrm{Oe}$, $M_s=14.46\times {10}^5\mathrm{A}∕\mathrm{m}$, $A=2.0\times {10}^{-11}\mathrm{J}∕\mathrm{m}$, and $\alpha =0.02$.

Figure 3

Maximum wall width narrowing $\Delta W∕W_0$ and the maximum angle $\phi$ as a function of the spin torque.

Figure 4

Snapshots of the moving magnetic wall at $t=0.06\mathrm{ns}$ in a Co nanowire. The damping constant $\alpha =0.1$ and the external field $H_{ext}=0\mathrm{Oe}$.

Figure 5

(a) The wall positions at several times $t=0$, $0.5\mathrm{ns}$, $1.0\mathrm{ns}$, $1.5\mathrm{ns}$, $2.0\mathrm{ns}$, $2.5\mathrm{ns}$, and $\mathrm{n}=3.0\mathrm{ns}$; (b) the wall velocity as a function of the wall position; (c) the maximum out-of-plane component of the magnetization as a function of the wall position. The damping constant is $\alpha =0.008$, the spin torque is $b_J=-750\mathrm{m}∕\mathrm{s}$, and the external field $H_{ext}=0\mathrm{Oe}$.

Figure 6

Domain-wall velocity as a function of time (a) for different spin torques at a fixed magnetic field $\left(10\mathrm{Oe}\right)$, and (b) for different magnetic fields at a fixed spin torque $\left(b_J=-500\mathrm{m}∕\mathrm{s}\right)$. The solid curves represent the analytical results and the scattered points are the numerical solutions.

Figure 7

Displacement of the domain wall for several spin torques without (a) and with (b) the external field. The solid curves represent the solutions of the trial function and the scattered points are the numerical results.

Figure 8

Energy power (a) and net energy gain (b) of the domain wall as a function of time.

Figure 9

Velocity $v$ and displacement $x$ for a pulsed spin current (left panels) and for a pulsed magnetic field (right panels). The damping constant is $\alpha =0.02$.

Figure 10

The phase boundary of pinning and depinning domain walls for two types of pinning centers; see text for the definition of the pinning parameters $V_0$ and $\zeta$. The external field is applied along the $-x$ direction and the spin current is along the $+x$ direction at $t=0$. The damping constant is $\alpha =0.02$.