Phenomenological description of the nonlocal magnetization relaxation in magnonics, spintronics, and domain-wall dynamics

A phenomenological equation called the Landau-Lifshitz-Baryakhtar (LLBar) [Zh. Eksp. Teor. Fiz 87, 1501 (1984) [Sov. Phys. JETP 60, 863 (1984)]] equation, which could be viewed as the combination of the Landau-Lifshitz (LL) equation and an extra “exchange-damping” term, was derived by Baryakhtar using Onsager's relations. We interpret the origin of this exchange damping as nonlocal damping by linking it to the spin current pumping. The LLBar equation is investigated numerically and analytically for the spin-wave decay and domain-wall motion. Our results show that the lifetime and propagation length of short-wavelength magnons in the presence of nonlocal damping could be much smaller than those given by the LL equation. Furthermore, we find that both the domain-wall mobility and the Walker breakdown field are strongly influenced by the nonlocal damping.

Figure 1

The spin-wave-amplitude decay along the rod, for a spin wave excited locally by applying a microwave $\mathbf{H}=H_{0}sin\left(2\pi ft\right){\mathbf{e}}_y$ of frequency $f=30\mathrm{GHz}$ and amplitude $H_0=1000\mathrm{Oe}$ in the region $0\le x\le 2\mathrm{nm}$. The data were fitted using Eq. (23) with $\beta =0.02$ and $\alpha =0.01$.

Figure 2

The spin-wave-decay constant–wave vector product $\lambda k$ as a function of the frequency for different $\beta$ values. The slate blue line was drawn using Eq. (29) for the case $\beta =0.01$.

Figure 3

Imaginary parts of the dynamical susceptibility ${\chi }_{yy}$ of the film for (a) thickness $d=300$ nm and (b) thickness $d=60$ nm.

Figure 4

Simulation results of domain-wall velocities for various susceptibilities. The parameters used are $\alpha =0.001, \beta =0, N_y=0.2$, and $N_z=0.8$. The vertical dashed lines are the breakdown fields computed using Eq. (53).

Figure 5

Simulation results of the magnetization length difference $\delta M$ for a 1D domain wall located at $x=500$ nm with $M_e=8.6\times {10}^5\mathrm{A}/\mathrm{m}$ and $A=1.3\times {10}^{-11}$ J/m. The demagnetizing factors are selected to be $N_x=0$ and $N_y=N_z=0.5$.

Figure 6

Simulation results of domain-wall velocities for the limit case that $\chi \to 0$ with various exchange dampings. The parameters used are $\alpha =0.005, N_y=0.4$, and $N_z=0.6$. The vertical dashed lines are the breakdown fields computed with Eq. (60).