Role of spin diffusion in current-induced domain wall motion for disordered ferromagnets

Current-induced spin transfer torque and magnetization dynamics in the presence of spin diffusion in disordered magnetic textures is studied theoretically. We demonstrate using tight-binding calculations that weak, spin-conserving impurity scattering dramatically enhances the nonadiabaticity. To further explore this mechanism, a phenomenological drift-diffusion model for incoherent spin transport is investigated. We show that incoherent spin diffusion indeed produces an additional spatially dependent torque of the form $\sim {\nabla }^2[\mathbf{m}\times (\mathbf{u}·\nabla )\mathbf{m}]+\xi {\nabla }^2\left[(\mathbf{u}·\nabla )\mathbf{m}\right]$, where $\mathbf{m}$ is the local magnetization direction, $\mathbf{u}$ is the direction of injected current, and $\xi$ is a parameter characterizing the spin dynamics (precession, dephasing, and spin-flip). This torque, which scales as the inverse square of the domain wall width, only weakly enhances the longitudinal velocity of a transverse domain wall but significantly enhances the transverse velocity of vortex walls. The spatial-dependent spin transfer torque uncovered in this study is expected to have significant impact on the current-driven motion of abrupt two-dimensional textures such as vortices, skyrmions, and merons.

Figure 1

Sketch of the one-dimensional Bloch wall implemented on a two-dimensional square lattice as used in the tight-binding model described in the main text.

Figure 2

Spatial distribution of the adiabatic and nonadiabatic torque for different current spin polarization. (a) $P=-1.0$, (b) $P=-0.56$, and (c) $P=-0.36$. The adiabatic torque remains essentially constant, while the nonadiabatic torque shows strong depends on the current spin polarization.

Figure 3

(a) Nonadiabaticity parameter as a function of the domain wall width in the absence (red) and presence of spin-conserving impurity scattering (black) of strength $V_{\mathrm{imp}}=0.1\mathrm{eV}$; (b) Non-adiabaticity parameter as a function of spin-conserving impurity scattering strength for a domain wall width of 8a. In these calculations, we set the polarization to $P=-0.56$. In the absence of disorder, the nonadiabaticity is governed by spin mistracking while it is dominated by dephasing when introducing disorder in the structure. The fitted parameters are ${\tilde{\beta }}_0=2.92,{\beta }_0=8.9\times {10}^{-3},{\beta }_0^{*}=4.27\times {10}^{-6},{\beta }_1=2.87\times {10}^{-4},{\beta }_2=5.24\times {10}^{-2},{\lambda }_L=0.97,{\tilde{\lambda }}_L=0.64$, and ${\lambda }_L^{*}=25$.

Figure 4

Spatial profile of the $y$ (a), (c) and $z$ components (b), (d) of the magnetization of a typical Bloch wall (a), (b) and vortex core (c), (d).

Figure 5

Normalized velocity of a one-dimensional Bloch wall, below $v_x^{<}$ (a) and above Walker breakdown $v_x^{>}$ (b). Numerical and analytic results are shown with dotted and solid lines respectively. The parameters used are $\alpha =0.005$ and ${\beta }_0=0.01,\chi =0.2$. (c) Spin-mistracking nonadiabaticity as a function of the transverse spin diffusion length ${\lambda }_{\mathrm{ex}}$ for decreasing domain wall widths, calculated based on Ref. [11].

Figure 6

Normalized transverse $v_y$ (a) and longitudinal $v_x$ (b) velocities of the vortex core. (c) Polar angle of the vortex core as a function of the core radius. Numerical and analytic results are shown with dotted and solid lines respectively. The parameters used are $R=150\mathrm{nm},\alpha =0.005,{\beta }_0=0.01$, and $\chi =0.2$.