Resonant translational, breathing, and twisting modes of transverse magnetic domain walls pinned at notches

We study resonant translational, breathing, and twisting modes of transverse magnetic domain walls pinned at notches in ferromagnetic nanostrips. We demonstrate that a mode's sensitivity to notches depends strongly on the mode's characteristics. For example, the frequencies of modes that involve lateral motion of the wall are the most sensitive to changes in the notch intrusion depth, especially at the narrow, more strongly confined end of the domain wall. In contrast, the breathing mode, whose dynamics are concentrated away from the notches is relatively insensitive to changes in the notches' sizes. We also demonstrate a sharp drop in the translational mode's frequency towards zero when approaching depinning which is confirmed, using a harmonic oscillator model, to be consistent with a reduction in the local slope of the notch-induced confining potential at its edge.

Figure 1

(a) Zero-field equilibrium magnetization configuration, ${\mathbf{m}}_0\left(\mathbf{r}\right)$, in a 75-nm-wide NiFe strip with symmetric notches ($w_{\mathrm{notch}}=20$ nm, $d_{\mathrm{notch}}=10$ nm) containing a head-to-head TDW with $m_y$ color scaling. The black arrows indicate the local magnetization direction. The $x$ and $y$ axis origins are also shown. (b)–(d) Snapshots of the translational, breathing, and twisting modes showing the dynamic component only $\left[\mathbf{dm}\right(\mathbf{r}\left)\right]$. The translational mode snapshot (b) uses $m_y$ color scaling and is taken when the TDW is displaced to the right ($+x$) at which point there is a significant dynamic $+m_x$ component. The breathing mode snapshot (c) also uses $m_y$ color scaling and is taken at the point during the TDW width oscillation when the width is larger than its equilibrium value. There is thus a large dynamic $+m_y$ component at the TDW edges which broadens the TDW. The twisting mode snapshot (d) uses $m_x$ color scaling and is taken at the point when the wide end of the TDW ($+y$) is displaced to the right and the narrow end of the TDW ($-y$) is displaced to the left (see also animations of the modes [57]).

Figure 2

(a), (b) TDW eigenfrequencies versus $d_{\mathrm{notch}}$ when varying $d_{\mathrm{notch}}$ for both notches simultaneously. (c), (d) Eigenfrequencies when varying $d_{\mathrm{notch}}$ only at one side of the strip, either at the wide end or narrow end of the wall while keeping the other notch with $d_{\mathrm{notch}}=10$ nm. For all data $w_{\mathrm{notch}}=20$.

Figure 3

Deformed domain wall in a 75-nm strip for $H_x=5530$ A/m.

Figure 4

Percentage change in $f_{\mathrm{breathe}}$ with respect to $f_{\mathrm{breathe}}$ at $d_{\mathrm{notch}}=10$ plotted against $d_{\mathrm{notch}}$ for (a) 5-nm-thick strips and (b) 2.5-nm-thick strips at various strip widths (see legends).

Figure 5

(a) Frequencies of the three TDW eigenmodes as a function of strip width, $w$. The notches are symmetric ($d_{\mathrm{notch}}=10$ nm, $w_{\mathrm{notch}}=20$ nm). At $w=88.4$ nm the calculated modes are hybrid breathing-twisting modes [see inset, (b)]. (c) shows snapshots of the amplitude of the dynamic component (red) of the hybrid modes found for $w=88.2$ nm at 6.091 GHz (upper, primarily a breathing mode) and 6.099 GHz (lower, primarily a twisting mode).

Figure 6

(a) $f_{\mathrm{trans}}$ versus in-plane field, $H$ (oriented along $+x$), for strip widths of 50, 60, 75, and 110 nm ($d_{\mathrm{notch}}=10$ nm and $w_{\mathrm{notch}}=20$ nm). (b) $f_{\mathrm{breathe}}$ and $f_{\mathrm{twist}}$ versus $H$ at a strip width of 75 nm. (c), (d) Snapshots of the amplitude of the dynamic component (red) of the magnetization for the (c) twisting and (d) breathing modes at a strip width of 75 nm for $H=5530$ A/m (i.e., close to depinning).

Figure 7

(a) Equilibrium TDW position versus $H$ applied along the $+x$ direction. Solid lines are linear fits to the low field data (typically the first four to five points). (b) TDW spring constant versus strip width calculated from the linear fits in (a) using Eq. (1). (c) Thiele domain wall width of the $H$-deformed TDWs versus $H$. (d) Calculated $f_{\mathrm{trans}}$ [calculated as per the text using the data in (a) and (c) and Eqs. (1, 2, 3)] versus the simulated $f_{\mathrm{trans}}$ [Fig. 6].

Figure 8

(a) $x$ dependence of the $x$ and $y$ components of the magnetization taken at $y=0$ (at the center of the strip). ${\mathrm{\Delta }}_T$ is the Thiele DW width and $\rho$ is a scaling factor used in the demagnetizing field calculation. (b) Effective width of the pinning potential ($L_{\mathrm{pin}}$) estimated from the maximum displacement of the TDW before depinning [taken from Fig. 7] plotted against ${\mathrm{\Delta }}_T$ for strip widths of 50, 60, 75, and 110 nm. The largest width strip has the largest ${\mathrm{\Delta }}_T$.

Figure 9

Plot of ${(dx/dH)}^{-1}$, proportional to the local effective spring constant, versus $f_{\mathrm{trans}}$ for field-displaced TDWs in strip widths of (a) 50, (b) 60, (c) 75, and (d) 110 nm. ${(dx/dH)}^{-1}$ and $f_{\mathrm{trans}}$ data were taken, respectively, from Figs. 7 and 6. Solid lines are linear fits to the data assuming a zero $x$-axis intercept. The inset in (d) shows the ratio of the slope of the data in (a)–(d) predicted from the spring model to the measured slope.

Figure 10

(a) $f_{\mathrm{twist}}$ versus the inverse strip width. (b) $f_{\mathrm{breathe}}$ versus $\sqrt{N_y}$ (see text for $N_y$ calculation) for a number of strip widths. The linear fits have been obtained by constraining the $x$-axis intercept to zero.