Adiabatic and nonadiabatic spin-transfer torques in the current-driven magnetic domain wall motion

A consistent theory to describe the correlated dynamics of quantum-mechanical itinerant spins and semiclassical local magnetization is given. We consider the itinerant spins as quantum-mechanical operators, whereas local moments are considered within classical Lagrangian formalism. By appropriately treating fluctuation space spanned by basis functions, including a zero-mode wave function, we construct coupled equations of motion for the collective coordinate of the center-of-mass motion and the localized zero-mode coordinate perpendicular to the domain wall plane. By solving them, we demonstrate that the correlated dynamics is understood through a hierarchy of two time scales: Boltzmann relaxation time ${\tau }_{\text{el}}$, when a nonadiabatic part of the spin-transfer torque appears, and Gilbert damping time ${\tau }_{\text{DW}}$, when adiabatic part comes up.

Figure 1

(Color online) (a) Stationary configuration of local spins $\left({\mathit{n}}_0\right)$ associated with a single Néel wall. Laboratory frames $x$, $y$, and $z$ and local frames $\overline{x}$, $\overline{y}$, and $z$ are indicated. (b) Schematic view of the TSA of itinerant spin $\mathit{s}$ and the OPZA of local spin $\mathit{n}$. These magnetic accumulations, respectively, cause the nonadiabatic torque ${\mathit{T}}_2$ and adiabatic torque ${\mathit{T}}_1$.

Figure 2

(a) Spatial profile of the polar angles $\phi \left(x,t\right)={\phi }_0\left[x-X\left(t\right)\right]$ and $\theta \left(x,t\right)=\pi /2+{\xi }_0\left(t\right){\Phi }_0\left[x-X\left(t\right)\right]$ in the flowing current state. (b) Linear dependence of ${\xi }_0$ and $\dot{X}\left(t\right)$ on the current density $j$. (c) Single-particle propagation (represented by solid line) with spin flip process by the $s\text{−}d$ interaction (represented by wavy line) which leads to the STT.

Figure 3

(Color online) The whole processes of establishing the nonadiabatic and adiabatic STTs. After switching on the electric field at $t=0$, the inequilibrium process toward the stationary flowing current state with time scale $t\simeq {\tau }_{\text{el}}$ (Boltzmann relaxation) causes the finite TSA $⟨{\overline{s}}_y⟩$ and resultant nonadiabatic torque, ${\mathcal{T}}_{\perp }$. Then, around the time scale of $t\simeq {\tau }_{\text{el}}+{\tau }_{\text{DW}}$ (Gilbert relaxation), the whole system reaches nonequilibrium but stationary state with the OPZA $\left(n_z\right)$ being established and macroscopic rotation of the DW spins being realized.

Figure 4

(Color online) (a) Walker’s choice of polar coordinates. (b) Profile of a moving domain wall and its projections onto the $xy$ plane (in-plane) and $xz$ plane (out-of-plane). When $\overline{\phi }={\overline{\phi }}_0=\text{small}$ constant, the profile on the $xz$ plane coincides with our quasizero-mode wave function.