Depinning of domain walls in permalloy nanowires with asymmetric notches

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Figure 2 show the magnetic hysteresis (M–H) loops obtained by FMOKE for the nanowires with d = 400 nm and 500 nm, respectively. The M–H loops for the other nanowires can be seen in Supplementary Material A. It needs to be emphasized that the M–H loop corresponds to the variation of the total magnetic moment of the nanowire versus the applied magnetic field (H_a), which is different from that obtained from a small local area at the nanowire in refs 11,22. The formation of the M–H loops shown in Fig. 2 can be understood as follows. First of all, the magnetic moments of the nanowires are saturated in the positive direction under H_a = 320 Oe. For the nanowire with d = 400 nm, as shown in Fig. 2(a), when H_a is decreased to zero and begins to increase in the negative direction, two jumps appear at −26 Oe and −56 Oe, which indicates respectively that a DW is nucleated at the elliptic pad and pinned at the notch after its injection into the left arm and propagation to the notch. When the magnitude of H_a is increased further until the nanowire is negatively saturated, another two jumps appear at −108 Oe and −226 Oe, which correspond to two distinctively different depinning fields of the DW. In contrast to the descending branch, there are only three jumps in the ascending branch. Apart from almost the same quantities of the two depinning fields (108 Oe and 229 Oe for the ascending branch), the DW nucleation and pinning processes seem to be overlapped, demonstrating only one jump at 38 Oe. For the nanowire with d = 500 nm, as shown in Fig. 2(b), there are two jumps in both the descending and ascending branches, which correspond to the nucleation (or pinning) field and the depinning field of DW. Similar two-jump phenomena can also be observed in the nanowire when d is increased from 600 nm to 1000 nm (see Supplementary Material A), which indicates that the appearance of one or two depinning fields depends strongly on d and will be discussed in the latter parts. As for the overlap of the nucleation field and the pinning field appearing in the descending branch and/or the ascending branch, it may be caused by the following two reasons. Firstly, from the simulation results (will be shown later), it can be seen that these two fields are approaching, which may make them not easily to be distinguished in experiments. Secondly, because of structural defects, edge/surface roughness and some other artificial effects, there may be some pinning sites in the nanowire, which results in some difference in the initial magnetization reversal process between the descending and ascending branches. However, because the depinning field is much larger than the nucleation (or pinning) field of DW, these artificial effects play little role and the quantities of the depinning fields are nearly the same in the descending and ascending branches.

Figure 2

The M–H loops obtained by FMOKE for the nanowires with.

(a) d = 400 nm and (b) d = 500 nm, respectively.

To confirm that the DW does exist in the nanowire and can be pinned at the notch, magnetic force microscopy (MFM) was operated at room temperature on the nanowire with d = 500 nm. To our knowledge, MFM can provide direct image of the stray field distribution along the nanowire with a fine tip magnetized perpendicular to the film plane^1,4,6,14,23. Figure 3(a,a’) show the MFM images for the nanowire at remanence state after a positive saturation field was applied and removed shortly. These two images were obtained respectively by two oppositely magnetized tips. Due to the roughness and defects, there still exists stray field caused by the net magnetic moments around the notch²³. In Fig. 3(a), the contrasts for the left-side and right-side of the notch are dark and bright, respectively²³. However, if the tip is magnetized oppositely, the relative contrasts around the notch are exactly reversed, as clearly shown in Fig. 3(a’). Moreover, the contrast of the right end is dark in Fig. 3(a) and switches to bright in Fig. 3(a’). All these results demonstrate that the nanowire resembles two bar magnets lined together with different poles on both sides of the notch²³, as indicated by the two blue arrows in Fig. 3(a,a’). According to the FMOKE results in Fig. 2(b) and the corresponding discussion, if a reversal field is applied at the plateau (between −70 Oe and −160 Oe), a DW will be pinned at the notch which is injected from the elliptic pad, and meantime the magnetic moments of the right arm have not reversed yet. To prove these statements by MFM, firstly the nanowire was saturated in the positive direction. After that, a reversal field of H_a = −150 Oe was applied and removed shortly. Then the MFM images were recorded as shown in Fig. 3(b,b’). Note that the magnetized states for the MFM tip in Fig. 3(a,b) (or Fig. 3(a’’)) are identical. From the results shown in Fig. 3(b,b’), the contrasts for the left side and the right side around the notch are nearly the same, which indicates that the magnetic moments of the left arm have reversed while those of the right arm remain unchanged. Therefore, now the nanowire resembles two bar magnets lined together with the same poles on both sides of the notch. In a word, the MFM images in Fig. 3 verify that a DW can be pinned around the notch by applying an external magnetic field properly.

In order to visualize the DW type and understand the detailed processes of DW pinning and depinning, micromagnetic simulation was performed. Note that the simulation was conducted on the nanowire identical to that in experiment (see details in the ‘Methods’ part), which favors to provide more reliable comparison between the simulated and experimental results. This is quite different from those done in refs 11,22, where the micromagnetic simulation is conducted on a part of the nanowire around the notch [11] or a much shortened nanowire [22] because the nanowires studied in experiments are too long (tens of or about 100 micronmeters in length). The detailed reversal process for the permalloy nanowire with d = 500 nm is simulated and exhibited in Fig. 4(a) with the magnitude of H_a increased from zero to 400 Oe which can fully saturate the nanowire’s magnetic moments. Several specific points marked as N, P, D in this plot represent the initial state of the DW nucleation, pining and depinning process, respectively. Moreover, P’ means the final state of the pinning process, after which the depinning process begins.

Figure 4(b) show the magnetic moment distribution corresponding to those specific points marked in Fig. 4(a), in which the arrows combined with the contrasts indicate the orientations of the magnetic moments along the nanowire. When H_a is zero, almost all the magnetic moments orient along the positive direction except that some portions are parallel to the edges due to strong shape anisotropy^3,21. When |H_a| is increased, the magnetic moments of the ellipse part begin to decrease with a first jump at H_a = −22.5 Oe, which means a CW VDW nucleating at point N (Picture N). Although the simulation results show that the probabilities of CW and CCW VDWs nucleated at the elliptic pad are equal, only CW VDW is taken as an example to be considered here and in the latter parts of the paper. When H_a reaches −47.5 Oe, the CW VDW is injected into the left arm and trapped at the notch^11,22,24 with a second jump accompanied by a significant reversal of the magnetic moments, which means the magnetic moments of the left arm have almost reversed to the negative direction while those of the right arm still keep unchanged (Picture P). Note that the difference between the nucleation and pinning fields is only 25 Oe, which may be too small to be distinguished by the M–H loop measurement. From the enlarged view of the notch area in Picture P, one can find that the magnetic moments near the left edge of the notch still point to the original direction due to strong shape anisotropy. Moreover, the CW VDW contracts and the core of the CW vortex moves to the upper corner of the notch left-side. If the applied field is removed now, the CW VDW will be expanded. To prove this statement, a simulation is performed at remanence state after H_a = −100 Oe is applied and then removed shortly, a larger CW VDW will be formed, as shown by the enlarged view in Fig. 4(c). This picture is in good agreement with the MFM images (Fig. 3(b,b’)) obtained at remanence state.

With further increase of |H_a| above 47.5 Oe, the CW VDW will move rightwards with size shrinkage, as displayed by the comparison between pictures P and P’. When H_a reaches −157.5 Oe, the total magnetic moments of the right arm reverse finally with the DW depinning (Picture D) and a third jump occurs in Fig. 4(a). This depinning field value is perfectly consistent with that (−160 Oe) observed by FMOKE as displayed in Fig. 2(b). In addition, for the nanowires with d increased from 500 nm to 1000 nm, this consistency is always kept well, which will be addressed in the latter parts. As for the inconsistency of the nucleation and pinning fields between the experimental and simulated results, it may be caused by some artificial effects as mentioned previously. Finally, as for the processes of DW reaching and passing through the notch, some transformation and even the chirality change of the DW may take place due to interacting with the notch (see details in Supplementary Material B), which depends strongly on d and will be discussed in the following.

In order to investigate the effect of the asymmetric notch shape on the pinning/depinning behavior of DW, further simulation was performed with increasing d from 200 nm to 1000 nm with an increment of 100 nm, which are the same as those done in experiments. To aid our understanding, the reversal process of the single right arm without the left part is also simulated. Figure 5(a) shows the d dependences of the nucleation field (H_N), pinning field (H_P), depinning field (H_D) of DW and reversal field of the single arm (H_R). Due to unchanged left part of the nanowire, it is reasonable that the values of H_N and H_P are kept respectively at −22.5 Oe and −47.5 Oe with varying d. In contrast, the magnitude of H_R increases quickly with decreasing d due to significant enhancement of shape anisotropy. The most important and interesting result in Fig. 5(a) is the d dependence of H_D. When d decreases gradually from 1000 nm, the CW DW maintain its chirality around the notch and meantime H_D increases slowly in magnitude and becomes stable at −157.5 Oe when d reaches about 700 nm. When d decreases further from 500 nm to above 200 nm (e.g. 400 nm), the DW either maintains or changes its chirality during passing through the notch, leading to two values of H_D, as indicated by the dash-lined box in the inset of Fig. 5(a). Finally, when d deceases to be very small (e.g. 200 nm), only CCW DW appears with one H_D again. The simulations for the same nanowire were repeated fifteen times to confirm the reliability of the results. For example, all the fifteen simulations show that only one chirality of the DW can exist at the notch for the nanowire with d = 500 nm. Finally, it needs to be noted that the values of the experimental H_D (indicated by ‘star’ dots) obtained by FMOKE are in good consistent with those of simulation when d decreases from 1000 nm to 500 nm. However, for the nanowire with d = 400 nm, the split of the two H_D values in experiment are much more significant than that in simulation. This may be due to structural defects, edge/surface roughness^25,26, which are not included in the micromagnetic simulation¹¹. When d is less than 400 nm, the depinning process can hardly be detected due to the sensitivity limitation of the FMOKE.