Remagnetization in arrays of ferromagnetic nanostripes with periodic and quasiperiodic order

We investigate experimentally and theoretically the magnetization reversal process in one-dimensional magnonic structures composed of permalloy nanostripes of the two different widths and finite length arranged in a periodic and quasiperiodic order. We showed that dipolar coupling between rectangular nanostripes is significantly reduced as compared to the analytical and numerical predictions, probably due to formation of the closure domains at the nanostripe ends. Although the main feature of the hysteresis loop is determined by different shape anisotropies of the component elements and the dipolar interactions between them, the quasiperiodic order influences the hysteresis loop by introducing additional tiny switching steps and change of the plateau width. We also showed that the dipolar interactions between nanostripes forming a ribbon can be counterintuitively decreased by reduction of the distance between the neighboring ribbons.

Figure 1

Section of a considered quasiperiodic structure. The flat and long magnetic NSs of thickness $t$, widths $w$ or $2w$, and length $L$ ($t\ll w\ll L$) are placed side to side and separated by air gaps of the width $s_x$. The chains of NSs form the ribbons separated from each other by the gaps of the width $s_y$. The green and red arrows show the exemplary direction of magnetization in narrow and wide NSs, respectively. The external magnetic field (black arrow) is placed along the NS axis.

Figure 2

The scanning electron microscopy images of a single ribbon formed by (a) periodic and (b) Fibonacci sequences of NSs.

Figure 3

Comparison of the hysteresis loops measured with L-MOKE in terms of (a) the separation between ribbons for the QPS with the NSs of $5\mu \mathrm{m}$ length and 50 nm thickness, and (b) for Fibonacci and PS array of NSs of $5\mu \mathrm{m}$ length, 50 nm thickness, and $10\mu \mathrm{m}$ separation between the ribbons. Vertical dashed lines in the inset mark the beginning and the end of the plateau. The labels I and II (III and IV) are related to switching of the wide (narrow) NSs. Scheme of the magnetic field lines from the wide NSs in the array when ribbons are well separated and close to each other are shown in (c) and (d), respectively.

Figure 4

Number of wide NSs (of 700 nm width), which are switched in successive intervals of external field for the Fibonacci QPS differing in the NS length / NS thickness / separation between ribbons: (a) $5\mu \mathrm{m}$/50 nm/$\infty$ (single ribbon), (b) $5\mu \mathrm{m}$/50 nm/760 nm, (c) $5\mu \mathrm{m}$/30 nm/760 nm, and (d) $10\mu \mathrm{m}$/30 nm/760 nm. NWN is a wide NS between two narrow NSs, and ${\mathrm{NWWN}}_1$ and ${\mathrm{NWWN}}_2$ is a wide NS from a pair of two wide ones between the narrow NSs which is switched in a pair at a lower or higher field, respectively (see also the legend at the top of the figure for color bar definition).

Figure 5

The structure field $H_{\text{str}}$ in the PS and QPS calculated for (a) a wide stripe (at NWN and NWWN positions) with P configuration of NSs, (b) a wide stripe (at NWN and NWWN positions) with AP configuration of NSs, and (c) a narrow stripe with AP configuration of NSs. The fields in (a), (b), and (c) are related to the points I, II, and III in the hysteresis loop marked in Fig. 3, respectively. The results for arrays of ribbons with the separation of 10 and $0.76\mu \mathrm{m}$ are marked with different colors. We plot $H_{\text{str}}$ along the NS axis (along $y$ axis) in the middle of the NS.

Figure 6

Dependence of the external magnetic field at the selected points of the experimental hysteresis loop (I, II, III, and IV) where switching of the selected NS happens on the respective structure field calculated from the analytical model. The plotted lines are calculated regression lines from which the values of $\kappa$ and $H_{\text{sw}}$ were extracted and collected in the Table 1. The results are shown for samples investigated experimentally in the paper.

Figure 7

Comparison of the hysteresis loops obtained from MC simulations. For (a) the single Fibonacci chain of magnetic moments corresponding to $5-\mu \mathrm{m}$-long and 50-nm-thick NSs for different values of the $k$ parameter and (b) for the periodic and Fibonacci sequence of the magnetic moments, corresponding to $5-\mu \mathrm{m}$-long and 50-nm-thick NSs at $k$ fixed to 0.1.

Figure 8

Results of the micromagnetic simulations. (a) Hysteresis loops for the single wide (700 nm) and narrow (350 nm) NS. The empty circles point at the fields for which the magnetization configuration is shown in (b)–(g). [(b)–(d)] Magnetization inside of the wide NS at (b) 0, (c) 130, and (d) 140 Oe. [(e)–(g)] The magnetization configuration inside a narrow NS at magnetic fields (e) 0, (f) 290, and (g) 300 Oe. The color indicates the orientation of the magnetization in the NS plane according with the color map shown in the inset. The intensity show the magnetization orientation in the plane perpendicular to the plane of NS, with the white color marked orientation perpendicular to the plane of the NS.