Domain-wall flexing instability and propagation in thin ferromagnetic films

We investigate field-driven domain-wall dynamics in thin ferromagnetic layers in the precessional regime using an analytical model that goes beyond the rigid wall approximation and takes into account the flexing modes of domain walls. This model allows us to turn on and off the stray field from domains adjacent to the domain wall in order to elucidate its role. We determine the eigenfrequencies of the flexing modes. The domain-wall flexing instabilities are shown to exist even without the stray field. The amplitude of the flexing modes shows a maximum when the magnetization precession frequency is equal to their eigenfrequency. At maximum amplitude of the flexing modes, the domain-wall velocity exhibits bumps even without the stray field. The stray field further enhances the velocity and broadens the bumps. This study is completed by micromagnetic simulations that show excellent agreement with the analytical model. The role of the various spatial modes in the velocity enhancement is clarified. Additional fractional frequencies appear in the time-dependent amplitude of the flexing modes, in analogy with the behavior of parametric oscillators.

Figure 1

(a) Schematic of the sample film (side view) and definition of the notations used in the text. (b) Profiles of the three first flexural modes of the DW.

Figure 2

Field dependence of the Floquet multiplier for mode 1 ($h_n=H_z/2\pi M$). Long dashed line $\alpha =0.1$, solid line $\alpha =0.2$, and short dashed line $\alpha =0.3$. The layer thickness is $d/\Lambda =4$. The arrows indicate the fields at which the precession frequency is equal to $f_1$ or $f_1/2$, $f_1$ being the eigenfrequency of the first flexural and precessional modes [Eq. (15)]. The inset shows the growth rate of mode 1 calculated from the parametric oscillator model [Eq. (19)] for $\alpha =0.2$ and $d/\lambda =4$.

Figure 3

Field dependence of the Floquet multiplier for mode 1 for different thicknesses of the film. Long dashed line $d/\Lambda =5$, dash-dotted line $d/\Lambda =4$, and solid line $d/\Lambda =3$. $\alpha =0.2$. The arrows indicate the fields at which the precession frequency is equal to $f_1$ or $f_1/2$, $f_1$ being the eigenfrequency of the first flexural and precessional modes [Eq. (15)].

Figure 4

The Floquet multiplier for the second flexing mode. $\alpha =0.1$, $d/\Lambda =4$.

Figure 5

DW velocity $v$ (averaged over film thickness and precession period) as a function of field strength. Calculation according to PDE [Eqs. (8)] with the stray field (solid circles) and without (open circles), calculation by the reduced set of equations [Eqs. (22, 23, 24, 25)] (triangles), 1D model (full curve). The dashed areas represent the instability regions. The arrows indicate the fields at which the precession frequency is equal to $f_1$ or $f_1/2$, $f_1$ being the eigenfrequency of the first flexural and precessional modes [Eq. (15)]. The parameters are $\alpha =0.2$, $d/\Lambda =4$, $\Delta /\Lambda =0.3$.

Figure 6

(a) DW velocity as a function of the applied magnetic field, 1D-model (red line), micromagnetic simulation (squares) using $d/\Lambda =4$, $\Delta /\Lambda =0.3$, and $\alpha =0.2$. (b) Amplitudes (rms values) of the first three flexural modes $q_1$, $q_2$, and $q_3$ of the DW.

Figure 7

(a) DW velocity as a function of the applied magnetic field, micromagnetic simulations (squares), calculation according to PDE [Eqs. (8)] with the stray field (solid circles), and without (open circles). The parameters are $d/\Lambda =4$, $\Delta /\Lambda =0.3$, and $\alpha =0.2$.

Figure 8

(a) Field dependence of the main frequencies in the Fourier spectrum of the amplitude of the first flexural mode $q_1\left(t\right)$ for $\alpha =0.2$ and $d/\Lambda =4$. The precession frequency $f_p=f_0/2$ and its overtones are represented by black symbols (the full and dotted lines represent the frequencies calculated within the 1D model). The subharmonic frequency $f_p/3$ and its overtones are represented by blue empty squares. (b) Amplitude of the two main frequencies $f_p$ and $f_p/3$ (in gray, the velocity curve) in the Fourier spectrum of $q_1\left(t\right)$.

Figure 9

(a) DW velocity as a function of the applied magnetic field for a damping coefficient $\alpha =0.1$ and $d/\Lambda =4$, micromagnetic simulation (squares), 1D model (red line). (b) Amplitudes (rms values) of the first three flexural modes of the DW $q_1$, $q_2$, and $q_3$.

Figure 10

Amplitude of the main frequencies in the Fourier spectrum of the first flexural mode $q_1\left(t\right)$ for a damping coefficient $\alpha =0.1$ and $d/\lambda =4$ (in gray, the velocity curve).