Resonance in magnetostatically coupled transverse domain walls

We have observed the eigenmodes of coupled transverse domain walls in a pair of ferromagnetic nanowires. Although the pair is coupled magnetostatically, its spectrum is determined by a combination of pinning by edge roughness and dipolar coupling of the two walls. Because the corresponding energy scales are comparable, the coupling can be observed only at the smallest wire separations. A model of the coupled wall dynamics reproduces the experiment quantitatively, allowing for comparisons with the estimated pinning and domain wall coupling energies. The results have significant implications for the dynamics of devices based on coupled domain walls.

Figure 1

(a) Scanning electron micrograph of a $w=85$ nm, $d=40$ nm device. (b) Micromagnetic simulation of two coupled DWs in a $w=100$ nm, $d=20$ nm pair of nanowires. Arrows indicate the in-plane direction of $\mathbf{M}$, and colors the magnitude of $M_x$ ($+M_s$, red; $-M_s$, blue). (c) 2D optical reflectivity map and (d) Kerr rotation map of a $w=85$ nm, $d=130$ nm device. The solid lines in (c) are the nanowire positions. The Kerr rotation map is shown for the maximum response on resonance.

Figure 2

(a) DW spectra for $w=85$ nm nanowires taken at two DW nucleation sites and in a separate device. Spectra are offset vertically for clarity. The solid lines are Lorentzian fits from which the central frequency is extracted. (b) Averaged pinning site frequency as a function of nanowire width for experiment (squares) and micromagnetic simulation (circles). Error bars indicate dispersion of pinned mode frequencies over the range of tested devices in experiment and simulation.

Figure 3

Separation dependence of (a) experimental and (b) simulated coupled DW resonances. Open symbols correspond to the lowest frequency resonance observed in each spectrum, closed symbols correspond to the next lowest frequency, and a cross designates the two third modes seen in experiment. Data is included for all nanowire widths. Hatched regions in (a) and (b) are the average plus or minus one standard deviation of the single nanowire resonances from Fig. 2. Inset in (a) is the spectrum for a $w=85$ nm, $d=130$ nm nanowire pair with two resonances at 1.5 and 2 GHz. In (b) the solid line is the simulated DW-DW resonance for the case of a $w=100$ nm pair of nanowires with zero edge roughness.

Figure 4

(a) Simplified model of the coupled DWs: two oscillating masses coupled together with spring constant $k$. (b) Energy profiles for the edge roughness of a $w=100$ nm nanowire (lower) and the DW interaction (upper) for a $w=100$ nm, $d=20$ nm pair, as a function of the position of the DWs in the nanowires, offset for clarity. Nanowires shown in the background share the $x$-axis scale. (c) Numerically solved $M_z^2={(sin{\varphi }_1+sin{\varphi }_2)}^2$ of the 1D coupled DW equations of motion [Eq. (1)] as a function of drive frequency and plotted for multiple nanowire separations, shown for the case of two $w=100$ nm nanowires. Values used were $k_1=0.21$ $\mathrm{erg}/{\text{cm}}^2,k_2=0.40$ $\mathrm{erg}/{\text{cm}}^2$, and $k=0.03\text{–}0.22$ $\mathrm{erg}/{\text{cm}}^2$. Insets in bottom left (upper right) are representations of the lower (higher) frequency mode.