Motion of Domain Walls and the Dynamics of Kinks in the Magnetic Peierls Potential

We study the dynamics of magnetic domain walls in the Peierls potential due to the discreteness of the crystal lattice. The propagation of a narrow domain wall (comparable to the lattice parameter) under the effect of a magnetic field proceeds through the formation of kinks in its profile. We predict that, despite the discreteness of the system, such kinks can behave like sine-Gordon solitons in thin films of materials such as yttrium iron garnets, and we derive general conditions for other materials. In our simulations, we also observe long-lived breathers. We provide analytical expressions for the effective mass and limiting velocity of the kink in excellent agreement with our numerical results.

Figure 1

(a) A Bloch domain wall with a kink in a thin film with perpendicular anisotropy. The center of the domain wall is indicated as a translucent strip. The segments of domain wall on either side of the kink lie in different valleys of the Peierls potential (top). (b)–(c) Side views of the domain wall corresponding to horizontal lines in (a), where the domain wall is at a minimum [(b), solid lines in (a)] or a maximum [(c), dashed line in (a)] of energy. While the continuum magnetization profile is the same for (b) and (c), the atomistic configuration is subtly different, which is the microscopic origin of the Peierls potential.

Figure 2

Simulations of kink–antikink collisions with initial velocities of $\pm 0.2Y/T$ ($\alpha =0.0004$, $l_1=1.58a$, $w\approx 79a$). (a) Initial configuration. We extract the domain-wall profile $x(y)$ (strip) from the atomistic simulations. Not all magnetic moments are shown. (b) For $w\lesssim l_2$, colliding kinks annihilate under emission of Winter spin waves [23]. (c) If conditions (11) and (14) are both satisfied, colliding kinks pass through each other as solitons. A segment of the domain wall makes a jump of distance $2a$ into another Peierls valley, and propagation continues. (d) For $\alpha w\gtrsim l_2$, colliding kinks lose energy through Gilbert damping. As in (b), they become trapped in a breather and eventually annihilate.

Figure 3

Domain-wall profiles $x(y,t)$ extracted from atomistic simulations of a breather ($w\approx 79a$, $\omega =0.25$). (a) If $w\sim l_2$, the solitonic (sine-Gordon) picture of DW kinks is inapplicable. The breather loses energy through spin-wave emission. (b)–(c) Spin-wave emission is virtually absent for $w\gg l_2$. However, for very high $w/l_2$ the breather is more susceptible to Gilbert damping ($\alpha =0.0004$), resulting in a faster-decreasing amplitude and period. (d) Solitonic limit ($w\gg l_2$ and no Gilbert damping). A video of the atomistic simulation is available [18].

Figure 4

(a) Our analytical expression (17) for the kink effective mass $m_{\mathrm{eff}}$ is in very good agreement with the numerical values for the atomistic model. Small corrections result from higher-order exchange terms [18]. For $w\lesssim l_2$, $m_{\mathrm{eff}}$ is reduced with respect to the sine-Gordon value $m_{\mathrm{sol}}$. (b) Final velocity $v_{\text{final}}$ for $\alpha =0.02$ and $\alpha =0.04$ and for three values of $w/l_2$ ($w\approx 79a$). In the regime $v_{\text{final}}\ll Y/T$, $v_{\text{final}}$ follows Eq. (20) (solid lines) and is independent of $w/l_2$. Deviations occur when $v_{\text{final}}$ becomes comparable to $Y/T$ (horizontal lines). Unlike in the sine-Gordon model, the kink velocity can exceed $Y/T$.