Influence of uniaxial anisotropy on domain wall motion driven by spin torque

Magnetization dynamics of a bilayer structure in the presence of a spin-transfer torque is studied using an atomistic model coupled with a model of spin accumulation. The spin-transfer torque is decomposed into two components: adiabatic and nonadiabatic torques, expressed in terms of the spin accumulation, which is introduced into the atomistic model as an additional field. The evolution of the magnetization and the spin accumulation are calculated self-consistently. We introduce a spin-polarized current into a material containing a domain wall whose width is varied by changing the anisotropy constant. It is found that the adiabatic spin torque tends to develop in the direction of the magnetization whereas the nonadiabatic spin torque arising from the mistracking of conduction electrons and local magnetization results in out-of-plane magnetization components. However, the adiabatic spin torque significantly dominates the dynamics of the magnetization. The total spin-transfer torque acting on the magnetization increases with the anisotropy constant due to the increasing magnetization gradient.

Figure 1

Schematic representation of the spin-transfer torque consisting of the adiabatic (AST) and nonadiabatic (NAST) torques in the rotated basis system.

Figure 2

The the tail-to-tail domain wall contained in the second ferromagnet of the bilayer system with the uniaxial anisotropy constant of $k_u=2.52\times {10}^6{\text{J/m}}^3$: The arrows indicate the direction of magnetization. The magnetization along the $y$ direction is represented by the blue coloring. In contrast, the red color shows the orientation of magnetization in the $-y$ direction.

Figure 3

Schematic representation of the magnetization component with time evolution from $0\mathrm{ns}$ to the equilibration time of $0.6\text{ns}$: The current density injected into the bilayer system containing the DW is $50 {\text{MA/cm}}^2$.

Figure 4

The time evolution of the spatial spin-transfer torque with $j_e=50{\text{MA/cm.}}^2$

Figure 5

(Top) The time variation of domain wall displacement with different current densities. (Bottom) The initial DW velocity as a function of current density: The critical current density, minimum current density required to move DW is $0.5{\text{MA/cm}}^2$.

Figure 6

The magnetization component of the initial DW center with time evolution after injecting the spin current with the density of $1000{\text{MA/cm}}^2$.

Figure 7

The domain wall profile transverse in the $xy$ plane with various anisotropy constants: The uniaxial anisotropy constant of cobalt is $K_u=4.2\times {10}^5{\text{J/m}}^3$. The distance between layer is given in units of supercells, corresponding to five atomic spacings.

Figure 8

The component of magnetization in the second FM with various anisotropy constants after the introduction of the spin current for $1 \text{ns}$: The center of the DWs are displaced in the direction of injected spin current. The system with high anisotropy constant leading to a large gradient of magnetization within domain wall results in a large displacement of the DW.

Figure 9

(top) The time-dependent variation of the domain wall displacement and (bottom) the initial domain wall velocity of different uniaxial anisotropy systems with the spin current density of $50{\text{MA/cm}}^2$.

Figure 10

The thickness dependence of a maximum of adiabatic spin torque (AST), nonadiabatic spin torque (NAST) and the degree of nonadiabaticity $\left(D_{\mathrm{NAST}}\right)$: The spin diffusion length $\left({\lambda }_{\mathrm{sdl}}\right)$ is $60 \text{nm}$.