Remarkably enhanced current-driven 360° domain wall motion in nanostripe by tuning in-plane biaxial anisotropy

By micromagnetic simulations, we study the current-driven 360° domain wall (360DW) motion in ferromagnetic nanostripe with an in-plane biaxial anisotropy. We observe the critical annihilation current of 360° domain wall can be enhanced through such a type of anisotropy, the reason of which is the suppression of out-of-plane magnetic moments generated simultaneously with domain-wall motion. In details, We have found that the domain-wall width is only related to K y − K x, with K x(y) the anisotropy constant in x(y) direction. Taking domain-wall width into consideration, a prior choice is to keep K y ≈ K x with large enough K. The mode of domain-wall motion has been investigated as well. The traveling-wave-motion region increases with K, while the average DW velocity is almost unchanged. Another noteworthy feature is that a Walker-breakdown-like motion exists before annihilation. In this region, though domain wall moves with an oscillating behavior, the average velocity does not reduce dramatically, but even rise again for a large K.

of magnetic moments easily lowers the energy of a kink, or equivalently, there is no topological object in 1d classical Heisenberg model 36,37 . Therefore, a suppression of out-of-plane magnetic moment by means of in-plane magnetic anisotropy may be feasible and necessary.
In this paper, we study the effects of a biaxial magnetic anisotropy on current-driven behavior of 360DW in a magnetic nanostripe by micromagnetic simulations. For simplicity, we consider a situation where two easy axes locate at x and y directions, respectively (see Fig. 1 for more details). We have found that the domain-wall width is only related to K y − K x . In principle, for a given K x , u c would always increase with K y . However, for K y > K x , domain-wall width is broadening considerably. Thus, a prior solution is to keep K y ≈ K x with a possibly largest K. We also observe two types of domain-wall motion before annihilation. In the range of small current, the average velocity of 360DW increases linearly with the current (traveling-wave motion) and is almost independent of the in-plane biaxial anisotropy. On the other side, in the range of large current, the displacement, time-dependent velocity and the out-of-plane magnetic moment of 360DW oscillate synchronously, exhibiting a Walker-breakdown-like behavior, while the structure of 360DW keeps stable. The average velocity of 360DW does not reduce dramatically, or even re-rise for a large enough K.

Model
As shown in Fig. 1, the magnetic nanostripe (Permalloy stripe) used in this paper is 4096 nm long in the x direction, 48 nm wide in the y direction and 5 nm thick in the z direction. For a thin enough film, the magnetization in the z direction should be uniform. Thus we can treat such a system as a two-dimensional one. In the initial state, a 360DW is placed in the center of the nanostripe. In such a magnetic nanostripe, there are sizeable distortions occurring in left-end and right-end edges of the stripe, which may somewhat affect the dynamics of the DWs. In order to avoid the distortions, the local magnetization in the left-end and right-end edges of the stripe are pinned in plane. The Hamiltonian of the stripe with magnetic anisotropy can be written as

Result
Biaxial anisotropy case with K x = K y . Based on a sequence of calculations, we find an effective way to enhance u c by taking both easy x-axis and y-axis anisotropy into account. Figure 2 shows average velocity 〈v〉 (defined by a convergent value of displacement over time) of 360DW as a function of spin current velocity u with different values of biaxial anisotropy, where K x = K y and K z = 0 have been set. As a benchmark, we plot u − 〈v〉 curve of TDW with K x = K y = K z = 0 in Fig. 2(a) as well (It should be noticed that, anisotropy can also enlarge the Walker breakdown current u w of TDW 40,41 ). One could immediately see that, without any uniaxial anisotropy, the 〈v〉 of 360DW soon annihilates thoroughly after a short linear increasing region. We then define the annihilation point as the critical current u c . Here, a vortex core emerges first and soon move out of the nanostripe. Thus the annihilation is irreversible. As a contrast, an antivortex appears in TDW case, which induces an oscillation in both the magnetic configuration and the velocity 18 . As u further increases, TDW would turn into a vortex wall, and one can observes the rapid increment of 〈v〉 for the vortex wall. Another distinction is u c is significantly smaller than the benchmark u w , which has also been reported before 18,26,33 .
In Fig. 2(b), we take K x = K y > 0 into account, one can see both u c and linear increasing region indeed increase with K x,(y) , and can easily go beyond u w . In addition, one can notice there are two or three regions of domain-wall motion before it annihilates, depending on the strength of anisotropy. Taking the curve K x = K y = 292 KJ/m 3 as an example, in the first region, i.e., u < 81 m/s, the average velocity of 360DW just increases linearly with the current. In fact, comparing to the other values of K x,(y) , the linear slope is almost unchanged. One then can deduce that biaxial anisotropy does not need any expense of DW velocity. When u is larger than 81 m/s, an oscillating region emerges, and 〈v〉 sightly drops. At last, only in the case of large anisotropy, as in this example, we observe an re-ascendance region just before annihilation. It is known that domain-wall velocity depends on both u and m z , where m z denotes the z-axis component of total magnetic moment of 360DW, which is calculated within the area of 360DW (ranges about 120 in x and 48 nm in y). Since u simply increases here, we plot the corresponding average magnetization in z direction 〈m z 〉 (defined by an average value of m z in the same time interval) to explore the possible reason behind this unique behavior. As displayed by the open orange square, one can see: 1) 〈m z 〉 simply increases in the first linearly increasing region; 2) in the sightly dropped region, the decline of 〈m z 〉 compete with the ascent of u; 3) in the last region, 〈m z 〉 increases again and leads to the second rise of 〈v〉.
To explore the characteristic of domain-wall motion further, we present the details of these three types of domain-wall motion in Fig. 3, adopting u = 53 m/s, 165 m/s and 245 m/s with K x = K y = 292 KJ/m 3 as three typical examples. In Fig. 3(a), we show the displacement of 360DW. In the case of u = 53 m/s, 360DW exhibits a stationary behavior and moves rigidly with a stationary velocity. Whereas for u = 165 m/s and 245 m/s, 360DW moves with a oscillation behavior. For a better sense of 360DW motion, we further present instant time-dependent velocity v and the z-axis component of the magnetic moment m z of 360DW as a function of time in Fig. 3(b,c), respectively. One can notice, for u = 53 m/s, both v and m z soon become constant. As a contrast, for the cases of u = 165 m/s and 245 m/s, v and m z show quasi-periodical oscillations, and the position of peaks and valleys match exactly to the displacement. The oscillation results in the poor linearity of 〈v〉 − u characteristic in the range of u > 81 m/s (shown in Fig. 2(b)), presenting a Walker-breakdown behavior. Interestingly, the average velocity of 360DW in large current does not reduce dramatically.
We also plot the domain-wall snapshots at the peaks and valleys. As shown in Fig. 3(d-g), one can see the 360DW expands and contracts quasi-periodically, however, keeps stable. When 360DW contracts, both m z and velocity increase. On the other hand, the expansion would oppose motion. As a result, one could observe a Walker-breakdown-like behavior. This mechanism is different to the stalled anti-vortex nucleation 42 or a complete suppression of the anti-vortex 14 in the TDW case.
Various combination of anisotropy. In order to further understand the effective enhancement of u c caused by the biaxial anisotropy with K x = K y , we then investigate the effects of various combination of biaxial anisotropy on u c for a comparison. We first start with the simplest case, i.e., uniaxial anisotropy.
In Fig. 4, we present the results with only x-direction anisotropy. One can see that in subpanel (a), u c decreases with the increasing of K x , and subpanels (b)-(d) show the local magnetization in x, y and z directions, respectively. From all plots of magnetization, one can see that domain-wall width shrinks (we define the domain-wall width as the region of M z ≠ 0 or M x ≠ M s ). However, the maximum of M x and M y almost not change, while the peak of M z increases. It suggests that the width of 360DW as well as M z plays an important role to u c . In the case of increasing K x , the width of 360DW should decrease. Then, the local magnetization vector near and at the area of 360DW should turn out of plane to reduce the exchange coupling energy. As a result, the M z of the 360DW increases, which deduces the contribution of demagnetization energy, leading to the decreasing of u c .   We now turn to the case with only K y . As shown in Fig. 5. One can observe that both u c and domain-wall width increases with the increasing of K y in Fig. 5(a-d), respectively. As a contrast to the result under only K x ≠ 0, the width of 360DW now increases with K y . Another characteristic is only the maximum of M z shown in Fig. 5(d)  decreases with K y , which supports the significance of M z . As K y increasing, the descend of M z mainly comes from the in-plane anisotropic field. The increasing of u c can be owed to the decreasing of the M z of 360DW, preserving the demagnetization energy. Even though the u c can be enhanced by tuning K y , it is not a good choice because the width of 360DW is significantly enhanced, which is not suitable for practical use.
Based on the results shown in Figs 4 and 5, one can naturally realize that u c may be effectively enhanced simultaneously with a fixed DW width by the combination of K x and K y . As shown in Fig. 6(a-d), u c indeed increases with equal biaxial anisotropy, i.e., K x = K y , and the width of 360DW almost remains unchanged due to the different effects of K x and K y on DW width. The increasing of u c is owed to the decreasing of the M z of 360DW, which is demonstrated clearly in Fig. 6(d).
In the Fig. 7, we plot the contour plot of domain-wall width against K x and K y . It can be clearly seen from the plot that the domain-wall width only depends on the value of K y − K x . As for K x > K y , i.e., the lower right part of Fig. 7, domain-wall width increases slowly with the value of K y − K x . One can see that from K x − K y = 125 KJ/m 3 to K x = K y , the width only changes from 72 to 155. However, as for K y > K x , i.e., the upper left part of Fig. 7, the width increases rapidly from 155 nm to 280 nm, corresponding K y = K x to K y − K x = 40 KJ/m 3 . Besides these regions, 360DW becomes unstable in our simulations, which corresponds to the white area in the contour plot.
We finally plot the contour plot of u c against K x and K y in the Fig. 8. From the contour plot, one can learn that in principle, u c would always increase with K y for a fixed K x . However, for K y > K x , domain-wall width is broadening considerably. Thus, a prior solution is to keep K y ≈ K x with a possibly largest K x .

Discussion
Though previous work mainly focuses on the theoretical simulation, we notice that materials with a biaxial anisotropy have already been realized experimentally, examples ranges from CoFe 2 O 4 films on (100) MgO substrates 34 to Fe films on BaTiO 3 substrates 35 . 360DW on those materials as well as related applications may need further study.