Spin-wave contributions to current-induced domain wall dynamics

We examine theoretically the role of spin waves on current-induced domain wall dynamics in a ferromagnetic wire. At room temperature, we find that an interaction between the domain wall and the spin waves appears when there is a finite difference between the domain wall velocity ${\dot{x}}_0$ and the spin current $u$. Three important consequences of this interaction are found. First, spin-wave emission leads to a Landau-type damping of the current-induced domain wall motion toward restoring the solution ${\dot{x}}_0=u$, where spin angular momentum is perfectly transferred from the conduction electrons to the domain wall. Second, the interaction leads to a modification of the domain wall width and mass, proportional to the kinetic energy of the domain wall. Third, the coupling by the electrical current between the domain wall and the spin waves leads to temperature-dependent effective wall mass.

Figure 1

(Color online) Geometry. The wire is along the $x$ axis and the static domain wall profile is in the $\left(x,y\right)$ plane because of a strong perpendicular anisotropy along $z$. The spherical polar coordinates are defined with respect to the $z$ direction.

Figure 2

(Color online) Wave functions $\sqrt{2N_{\text{dw}}}{\xi }_{\text{loc}}\left(\mathit{r}\right)$ and $\sqrt{{\omega }_{\mathit{k}}N}{\xi }_{\mathit{k}}\left(\mathit{r}\right)$ about the domain wall at $x_0$ for $k_x=1/2\lambda$ and $k_x=4/\lambda$.

Figure 3

(Color online) (a) $\sqrt{N}{v}_{\mathit{k}}$ as a function of $k_x$ and $k_y$. (b) $L{v}_{\mathit{km}}$ as a function of $k_x$ and $m_x$. $K_{\perp }/K_u=57$ for both graphs.

Figure 4

(Color online) Dispersion relation of the magnons as a function of the spin-current velocity $u$ for $K_{\perp }/K_u=57$ and $\hslash /K_u=1.88\text{ns}$.

Figure 5

(Color online) (a) Correlation function $I\left(t-t^{\prime }\right)$ between the stochastic torque $T_{\text{stoc}}\left(t\right)$ at two different times $t$ and $t^{\prime }$ for a wire of Permalloy (Ref. 18) with $K_{\perp }/K_u=57$ and $\hslash /K_u=1.88\text{ns}$. (b) $I\left(t-t^{\prime }\right)$ for a wire of Co/Pt (Ref. 51) with $K_{\perp }/K_u=0.73$ and $\hslash /K_u=0.013\text{ns}$.

Figure 6

(Color online) Modification of the domain wall mass $m$ and of the domain wall width $\lambda$ as a function of the relative velocity $\left|u-{\dot{x}}_0\right|$ and the exchange stiffness constant $A$.

Figure 7

Representation of the force $F_j$ created by the spin waves when the relative velocity $u-{\dot{x}}_0$ is finite and the motion is adiabatic. The wavy line represents the magnon $q$ exchanged with the domain wall, the vertex × arises from the force $F_j$ and the vertex ● represents the potential $\mathcal{V}$.

Figure 8

(Color online) Modification of the domain wall mass $\delta m/m$ as a function of the wire height $h$, for different temperatures $T$, assuming $L_y=100\text{nm}$ and $\kappa =57$.

Figure 9

Representation of the force $F_H$ created by the spin waves when the excitation of the spin waves by the spin current is not adiabatic. The wavy line represents the momentum $\mathit{q}$ exchanged with the domain wall and the energy $\Omega$ transferred by the spin current; the vertex × arises from the force $F_H$ and the vertex ● represents the mixing potential $\mathcal{V}$.

Figure 10

(Color online) Contribution of the transverse spin-wave modes $n_y$ to the change in the domain wall mass $\delta m/m$. $L_y=400\text{nm}$, $\alpha =0.01$, and $\kappa =K_{\perp }/K_u=57$.

Figure 11

(Color online) Proposed experiment for spin-wave emission. A dc power supply $V_{\text{dc}}$ generates a charge current $j_c$, which in turn excites a domain wall pinned inside a wire. As the spin current $u$ is finite but at the same time the domain wall is fixed, the relative velocity $u-{\dot{x}}_0$ is finite and spin waves are emitted.

Figure 12

(Color online) (a) Odd part $\left(v_{\mathit{km}}-v_{\mathit{mk}}\right)/2$ of coefficient $v_{\mathit{km}}$. (b) Even part $\left(v_{\mathit{km}}+v_{\mathit{mk}}\right)/2$ of coefficient $v_{\mathit{km}}$.