Current-driven domain wall dynamics in ferromagnetic layers synthetically exchange-coupled by a spacer: A micromagnetic study

The current-driven domain wall motion along two exchange-coupled ferromagnetic layers with perpendicular anisotropy is studied by means of micromagnetic simulations and compared to the conventional case of a single ferromagnetic layer. Our results, where only the lower ferromagnetic layer is subjected to the interfacial Dzyaloshinskii-Moriya interaction and to the spin Hall effect, indicate that the domain walls can be synchronously driven in the presence of a strong interlayer exchange coupling, and that the velocity is significantly enhanced due to the antiferromagnetic exchange coupling as compared with the single-layer case. On the contrary, when the coupling is of ferromagnetic nature, the velocity is reduced. We provide a full micromagnetic characterization of the current-driven motion in these multilayers, both in the absence and in the presence of longitudinal fields, and the results are explained based on a one-dimensional model. The interfacial Dzyaloshinskii-Moriya interaction, only necessary in this lower layer, gives the required chirality to the magnetization textures, while the interlayer exchange coupling favors the synchronous movement of the coupled walls by a dragging mechanism, without significant tilting of the domain wall plane. Finally, the domain wall dynamics along curved strips is also evaluated. These results indicate that the antiferromagnetic coupling between the ferromagnetic layers mitigates the tilting of the walls, which suggest these systems to achieve efficient and highly packed displacement of trains of walls for spintronics devices. A study, taking into account defects and thermal fluctuations, allows to analyze the validity range of these claims.


INTRODUCTION
Understanding and controlling the dynamics of domain walls (DWs) along ultrathin magnetic heterostructures consisting of a ferromagnetic (FM) strip sandwiched between a heavy metal (HM) and an Oxide is nowadays the focus of intense research 1,2,3,4,5,6,7,8 . These been also carried out. In order to collect detailed information about the acting mechanisms associated to the coupling between the FM layers through the spacer, which is determined by a certain interlayer exchange parameter 16,17 , the magnetization state of both FM layers is simultaneously evaluated. Finally, both ferromagnetic (FM) and antiferromagnetic (AF) coupling cases are considered, the former given by a positive , and the latter by a negative . Figure 1. (a) Schematic representation of the multilayer structure with two FM layers. The relevant thicknesses for this study are marked on the figure, which are fixed to = = = 0.8 nm , except otherwise indicated. The saturation magnetizations are also fixed by default to = 600 kA/m and = 600 kA/m. The width is = 100 nm. The anisotropy constant, the intralayer exchange constant and the Gilbert damping are respectively 1 = 0.6 MJ/m 3 , = 20 pJ/m and = 0.1 for both FM layers. The interfacial DMI in the lower FM is = 1.25 mJ/m 2 , and no DMI is considered for the UFM ( = 0). Along this work, the single-FM-layer (b) is considered to have identical characteristics to those of the lower FM layer. Parameters here used can be found in the literature 4,7,14. The manuscript is structured as follows. Section 2 describes the details of the micromagnetic model ( ) and the one-dimensional model (1DM). The current-driven DW dynamics along perfect samples, both in the absence and in the presence of in-plane longitudinal fields, is presented in Section 3. Micromagnetic results for realistic samples are presented in Section 4 for different multilayers where the thickness of the FM layers and the spacer ( , and ) and the saturation magnetization of the layers ( and ) are varied. The current-driven DW motion along multilayers with curved parts is studied in Section 5, and the main conclusions are discussed in Section 6.
⃗⃗⃗( ⃗, ) ≡ ⃗⃗⃗ ( ⃗, ) = ⃗⃗⃗ ( ⃗, )/ is the normalized local magnetization to its saturation value ( ), defined differently for each FM layer: where : , for the LFM and the UFM layers respectively. ⃗ ⃗⃗ is the deterministic effective field, which includes not only the intralayer exchange and the uniaxial anisotropy, but also the interlayer exchange 17  The effective field in the LFM layer requires an additional term representing the interfacial DMI at the HM/LFM interface. The rest of numerical details of other contributions to the effective field can be found elsewhere 18 . ⃗ ⃗⃗ ℎ is the thermal field, included as a Gaussian-distributed random field 20,21 . ⃗ represents the spin-orbit torque ( Sections 4 and 5). In order to take into account the effects of disorder due to imperfections and defects in a realistic way, we assume that the easy axis anisotropy direction is distributed among a length scale defined by a 'grain size'. The grains vary in size taking an average size of 10 nm. The direction of the uniaxial anisotropy of each grain is mainly directed along the perpendicular direction ( -axis) but with a small in-plane component which is randomly generated over the grains. The maximum percentage of the in-plane component of the uniaxial anisotropy unit vector is varied from 10% to 15%. The presented results correspond to an in-plane maximum deviation from the out-of-plane direction of 12%.
Although other ways to account for imperfection could be adopted, we selected this one based on previous studies, which properly describe other experimental observations 19 .

2.2.One-dimensional Model (1DM)
The one dimensional model (1DM) assumes that the DW profile can be described by the Bloch's ansatz 9 , and therefore its dynamics can be described by means of the DW position ( ) and the internal DW angle (Φ). The 1DM has been developed by several authors to account for and describe the field-driven and current-driven DW dynamics in different systems 4,9,18 . Yang et al. 14

3.1.A Current-driven DW motion in the absence of longitudinal fields
We firstly describe the current-driven DW motion along perfect and straight systems.
Representative snapshots of the local magnetization before and just at the end of a 2-ns long current pulse with amplitude = = +2.5 TA/m 2 , are depicted in Fig. 2. In what follows, the units of the current density are given in TA/m 2 indicating 10 12 A/m 2 . Fig. 2(b) shows the results for a single DW in the single-FM-layer stack. Fig. 2 Interestingly, no DW tilting is observed for this AF coupling case.
where Φ is the terminal DW angle (Φ̇= 0, i.e. Φ → Φ ). Eq. (7) indicates that DW terminal velocity for a single-FM-layer monotonously increases with , but with a decreasing slope as the absolute current is increased 9,18,19 . This fact can be explained by the relative orientation of the magnetization within the DWs (⃗⃗⃗ or Φ with : , ) and the direction of the electric current flow ( ⃗ ( ) = ( )⃗⃗ ). In fact, the closer the direction of the magnetization within the DWs is to the direction of the current flow, the more efficient is the SHE pushing the DWs 3,18,26 . However, the spin orbit torque (SOT) due to the SHE itself promotes the progressive misalignment of both magnetization and current: as increases, the angle Φ asymptotically tends to 90 0 (see "DW angle vs " graph in Fig. 3(b)), leading to the abovementioned decrease of the slope of the DW speed dependence on current amplitude 18,19 .
Therefore angles (see "DW angles vs " graph in Fig. 3(c)), and the SOT is not sufficiently intense to promote a significant misalignment between the current flow and the magnetization within the DWs. Actually, the use of Eq. (8) derived from the 1DM, yields a rather good approach to compute this terminal DW velocity, provided the DW angle in the LFM layer is set to Φ ≈ 180 0 , as it can be seen from full simulations (see "DW angles vs " graph in Fig.   3(c)). However, a slight but progressive slope reduction is obtained as the current is increased, which can be ascribed to the increasing misalignment between the magnetizations within the paired DW, as the same graph reveals for high currents. Another important characteristic of this CDDW dynamics is that DW tilting completely vanishes, so that DWs hold perpendicular to the longitudinal direction ( -axis). We have shown that the micromagnetic ( ) results of the DW velocity vs can be qualitatively described by the analytical Eqs. (7) and (8)   The dependence on the applied current of the terminal values of the DW velocity and the DW angle is shown from top to bottom graphs. The parameters are those given in the text. The DW width for the AF coupling ( < 0) was needed to be rescaled to Δ ≈ 2.3 nm in the 1DM in order achieve quantitative agreement with results. For the two other cases, FM coupling ( > 0) and single-FM layer, the input value of the DW width was Δ ≈ 7.3 nm, as predicted by the analytical formula Δ = √ / . Perfect samples and zero temperature conditions are considered. Blue lines in (b) correspond to the 1DM results obtained taking into account the DW tilting (see Ref. 18 for details). The inset in the bottom graph of (b) represents the DW tilting as a function .
As it shown in Fig. 4, the 1DM predictions are in good qualitative agreement with the results, both for the DW velocity and the DW angles. The discrepancies between the and the 1DM results in the single FM layer (Fig. 4(b)) can be attributed to the approximated description provided by the 1DM, which neglects, among other aspects (such as the approximated description of the shape anisotropy field, for instance), the DW tilting observed in the results. Indeed, we notice that taking into account the DW tilting in the 1DM (see . It has to be also noticed that by imposing Δ ≈ 7.3 nm as the input for AF coupling case, the DW velocity predicted by the 1DM overestimates the results of "DW velocity vs " by a factor of ~3 (not shown), whereas the dependence of "DW angle vs " is hardly affected. For these reasons, we will continue analyzing in the following sections the current-driven DW dynamics along multilayers with two FM layers adopting a full micromagnetic description, which naturally accounts for the 3D dependence of the magnetization, including the magnetostatic interaction between them and the eventual DW tilting.

3.1.B The influence of the Spin Transfer Torques on the current-driven DW motion
In previous discussion we have assumed that the most of the current flows along the HM, so the only driving force on the DWs is due to the spin Hall effect (SHE), which drives DWs along the current direction for the chiral DW nature consider in the present study (lefthanded chirality imposed by the DMI). However, the current could also partially flow along the FM layers, and consequently the conventional adiabatic and non-adiabatic spin transfer torques (STTs) could also contribute to the current-driven DW dynamics. In order to explore the influence of these STTs, we have evaluated the DW dynamics along the same systems  The STT pushes the DW along the electron flow (against the current). As commented, for the single-FM-layer stack (Fig. 5(a)), the DW velocity due to the SHE (which drives the DW along the current direction) increases monotonously up to asymptotic saturation with in the absence of STTs ( = 0, black dots in Fig. 5(a)). When the STT is taken into account, the DW velocity decreases for a given current. This velocity reduction is larger as the nonadiabatic parameters increases from = 0 (open red symbols in Fig. 5(a)) to = 2 = 0.2 (open blue symbols in Fig. 5(b)). Fig. 5(b) also shows that the DW velocity reaches a maximum for a given current, and for large currents the DW velocity starts to decrease again.
These results indicate that the STTs act against the SHE, reducing the magnitude of the DW velocity, which is along the current direction. to play a marginal role for multilayers with AF coupling (Fig. 5(b)). On the other hand, we have also considered that the same current flowing through the HM is also flowing through the ultrathin FM layers. This is surely an exaggeration, as the electrical resistivity of the FM should be larger than the one of the FM layers 29 (2017)). In a more realistic case, the density current along the FM layer must be smaller than the one along the thicker and low-resistivity HM, and consequently the STT should play a marginal role. For these reasons, we will not take into account the STTs in the rest of the manuscript.

3.1.C Inertia effect on the current-driven DW motion
In Sec. 3.1A and 3.1B, we have plotted the terminal DW velocity reached by the DWs after application of constant density current. Such values where obtained at = 2 ns, which was found sufficient to achieve the steady-state terminal DW velocity. It is also interesting to evaluate the DW dynamics once the current pulse is turned off. The current-driven DW dynamics due to their own inertia has been studied in systems with in-plane magnetization by Thomas et al. 31 and Chauleau et al. 32 , where the DW motion when the current pulse was turned off was essentially ascribed to the gyrotropic dynamics of the vortex DW configurations. Vogel et al. 33 shown that the DW motion induced by nanosecond current pulses in Pt/Co/AlOx multilayers with perpendicular magnetic anisotropy exhibits negligible inertia. More recent studies by Torrejon et al. 19 have shown that inertia effects result in a DW motion even when the current is switched off in high PMA systems with low damping. Our aim here is just to evaluate the inertia in HM/LFM/Spacer/UFM stacks, with FM ( > 0) and AF ( < 0) coupling, and to compare this "after-effect" to the single-FM-layer stack.
To do it, we applied the current pulse at = 0, and monitor the temporal evolution of the DW position and the DW velocity along perfect samples (without disorder) and at zero temperature. The results are shown in Fig. 6

(a) and (b). For a single-FM-layer (open circles
in Fig. 6(a) and (b)), the DW takes some time to reach its terminal velocity from = 0. It also takes some time to reduce its velocity to zero once the current pulse is switched off at = = 2 ns. As expected, these acceleration and deceleration times increase for the FM coupling case (black squares in Fig. 6(a) and (b)) with respect the single FM layer stack. This is due to the larger effective DW mass of the FM coupled system as compared to the single-FM-layer stack 19 . Interestingly, the acceleration and deceleration times are significantly short for the system with AF coupling (blue triangles in Fig. 6(a) and (b)), which constitutes an additional advantage of these systems for some applications: DWs in these AF coupled stacks can be accelerated and decelerated faster as their single-FM-layer and FM coupled counterparts. Figure 6. results of the temporal evolution of the DW position (a) and the DW velocity (b) under a current pulse of = 2 TA/m 2 and = 2 ns. for the three evaluated cases: FM coupling ( > 0), single-FM layer, and AF coupling ( < 0). The same parameters as in Fig. 2 and 3 are considered. Depicted results correspond to perfect samples at zero temperature.

3.2.Current-driven DW motion under longitudinal fields
Other revealing study of the consequences of the AF coupling between the two FM layers is the dependence of DW motion on the application of an in-plane longitudinal field This behavior is qualitatively similar to that of a DW in a single-FM-layer ( Fig. 7(b)).
Differently from this behavior, an absolute decrease of the DW velocity is obtained under the application of the longitudinal field for the AF-coupled system ( < 0). As it has been shown, in the absence of driving force ( = 0), the magnetizations within the coupled DWs of the LFM and the UFM layers tend to be aligned antiparallel along the -axis (Φ ≈ 180 0 and Φ ≈ 0 0 ). The longitudinal field promotes the progressive misalignment with respect to -axis, independently of its sign. Therefore, due to the reduced SOT efficiency for such a misalignment, the velocity decreases as | | increases. In general, it can be observed that the DW tilting is not null in the presence of in-plane fields (see graphs in Fig. 7(a),(b) and (c)). Additionally, the DW width does not remain constant under (see bottom graphs in Fig. 7). We have also evaluated the 1DM predictions for the current-driven DW motion in the presence of longitudinal fields. The 1DM results are collected and compared to the results in Fig. 8. A good qualitative agreement is achieved for the three cases. The quantitative discrepancies are due to the same limitations discussed above for the pure current-driven case: the 1DM does not take into account the DW tilting angle nor the magnetostatic interaction between the two FM layers. Moreover, it assumes that the DW width is fixed, which is not the case of the full results shown in the bottom graphs of Fig.   7. Nevertheless, the 1DM gives a good description of the results provided that the DW width (Δ = 2.3 nm) is properly selected for the AF coupling case. The agreement is also good for the FM coupling and single-FM-layer cases adopting a constant DW width as deduced from the analytical formula Δ = √ / = 7.3 nm. . The dependence on of the DW velocity and the DW angle is shown from top to bottom graphs. The parameters are those given in the text. The amplitude and the duration of the current pulse are = 2.5 TA/m 2 and = 2 ns respectively, and the presented results were computed at = 2 ns. The DW width for the AF coupling ( < 0) was needed to be rescaled to Δ ≈ 2.3 nm in the 1DM in order achieve quantitative agreement with results. For the two other cases, FM coupling ( > 0) and single-FM layer, the input value of the DW width was Δ ≈ 7.3 nm, as predicted by the analytical formula Δ = √ / . Results correspond to perfect samples and zero temperature.

3.3.Current-driven DW motion as a function of the interlayer exchange coupling
Before discussing the case of realistic samples with imperfections, it is interesting to examine the current-driven DW dynamics for different values of the exchange coupling between the layers ( ). The results of the DW velocities of the lower and the upper FM layers are shown in Fig. 9 for two different current density amplitudes , and for two different  In this section, we study the current-driven DW motion along realistic strips, i.e. with imperfections (details were given at the end of Section 2.1), and considering different FM layers, with different thicknesses ( and ) and saturations magnetization ( and ). The thickness of the spacer is also varied ( ). Besides the FM coupling ( > 0 ), single-FM-layer and AF coupling ( < 0 ) cases, the results collected in Fig. 10 also include the case where the two FM layers are not exchange coupled ( = 0, red circles). Note that in the absence of interlayer exchange coupling, only the DW in the LFM is displaced as due to the SHE, which, as already mentioned, in the present work is only acting in the LFM layer. Therefore, red circles in Fig. 10 correspond to the DW velocity in the LFM layer, whereas black squares (FM coupling) and blue triangles (AF coupling) represent the DW velocities in both the LFM and the UFM layers, where they move coupled. Figure 10. results of the current driven DW motion along realistic strips for different combinations of the saturation magnetization in the FM layers (a,b,c, from top to bottom) and different thicknesses of the layers (1,2,3, from left to right). The magnitude of the exchange coupling parameter is | | = 0.5 mJ/m 2 for the FM ( > 0) and AF ( < 0) coupling cases and zero for the no coupling case ( = 0). These results were obtained at zero temperature for realistic samples, with defects included as described in Sec. 2.
Several important conclusions can be extracted from the results shown in Fig. 10.

CURRENT-DRIVEN MOTION ALONG CURVED STRIPS
Apart from the larger velocity of the DWs, another important advantage of using AF coupled layers with respect to the single-FM-layer stacks is the absence of DW tilting (see Fig. 2 and 3). In a single-FM-layer stack, adjacent DWs depict opposite tilting of their DW plane 34,35 , which imposes a limit in the density of information coded between adjacent DWs.  The current density ⃗ ( ⃗) becomes non-uniform when is forced to flow along curved paths. The current distribution in the HM under the lower FM layer is shown in Fig. 11(b), which clearly indicates a radial dependence: the current density ⃗ ( ⃗) depicts an inversely linear dependence on the radius when is forced to flow over semicircular arcs. These results were computed with COMSOL 22 and taken into account to evaluate the current driven DW motion. Realistic conditions have been considered, which include defects in the form of grains (see details at the end of Section 2.1) and thermal effects at room temperature ( = 300 K). Two cases are considered, a single-FM-layer stack (Fig. 12) and a multilayer with AF coupling (Fig. 13). In both cases, a series of DWs is initially placed at one of their ends   In principle, besides of the high efficient DW dynamics, AF-coupling systems can also improve the density of packed information, coded between adjacent walls. However, a deeper observation of the images shown in Fig. 13 indicates that the second down domain in UFM (second up domain in the LFM) is contracted when arriving at the first curve (third image in Fig. 13, at = 1 ns). Then, this down domain extends a little bit on the straight line and again contracts at the second curve. Therefore, it seems that under realistic conditions (defects and thermal fluctuations) the distance between adjacent DWs can also vary during the motion even for the AF coupling case, and consequently it is needed to evaluate the distance between adjacent DWs for realistic conditions. In order to get further insight into this behavior, we have also evaluated the dynamics of two DWs within each FM layer starting from different distances between them ( 0 ). We monitor the evolution of the distance between these DWs at five different points along the curved sample. These points are labeled with letters in Fig. 14(a), which corresponds to an initial state where two DWs initially separated by 0 ≈ 130 nm. The snapshots shown in Fig. 14(a) were obtained in the presence of disorder (see disorder details in Sec. 2.1) but at zero temperature ( = 0). It can be visually checked that the initial distance between the 2 DWs is not changing as they are driven along the track (see also Fig. 15(a)). However, in the presence of thermal noise at = 300K, we notice that the distance between the DWs slightly changes: see Fig. 14 Δ constitutes a measure of the distance between the DWs as they are driven along track.
As it is shown in Fig. 15(d), Δ does not vary from point to point at zero temperature (blue dots in Fig. 15(d)).
In order to provide a statistically description of this thermally activated dynamics, we evaluated three different grain patterns and three different stochastic realizations of the thermal noise at = 300K. The corresponding results of Δ at the mentioned points are shown by open symbols in Fig. 15(d), which indicates that the distance between the walls changes for different grains patterns and temperature realizations. However, the mean distance averaged over these grains patterns and stochastic realizations (red squares in Fig.   15(d)) is hardly dependent on the point along the track. We have performed a similar study starting from two DWs initially separated by 0 ≈ 60 nm, and we verified that the DWs can collapse for some of the evaluated stochastic realizations. Therefore, this imposes a limit in the density of packed information even for the AF coupling stacks. Although further studies are needed to evaluate other samples with different strip width and curvature radius, our analysis suggests that the AF coupled multilayers could be used to efficiently drive trains of highly packed DWs.

CONCLUSIONS
The current-driven DW motion has been studied by micromagnetic simulations in multilayers with two ferromagnetic layers separated by a spacer. These layers are coupled by the interlayer exchange coupling, which depending on its magnitude and sign, can generate ferromagnetic (FM) or antiferromagnetic (AF) coupling between them. The interfacial Dzyalozinskii-Moriya interaction is only required at the interface between the heavy metal layer and the lower ferromagnetic layer, and provides the magnetization domain wall texture with the adequate chirality. The results are compared to the ones obtained for the singleferromagnetic-layer case and qualitatively explained in terms of analytical expressions deduced from the one-dimensional model. However, the three-dimensional space micromagnetic description allows for unraveling some details of such dynamics that are not fully accessible from a one-dimensional description, even though the latter approach may draw rather good qualitative results. For low currents in perfect samples, the driving force resulting from spin-orbit torques (spin Hall effect) is not capable to impel paired walls as efficiently as domain walls in the single-ferromagnetic-layer stack. Indeed, domain walls in the upper ferromagnetic layer are dragged by the moving walls in the lower ferromagnetic layer, because of the interlayer exchange coupling, which results in this lack of effectiveness.
For higher currents, the coupled walls associated to the FM coupling present an analogous behavior to that of domain walls in the single layer stack, i.e., the domain wall velocity saturates as the current is increased.
On the other hand, the AF coupling results in a high velocity of the coupled DWs, which are driven without significant tilting by the spin Hall effect from the heavy metal. The antiferromagnetic coupling promotes the antiparallel alignment of the internal DW moments in the lower and in the upper layers, both depicting a chiral Néel configuration. As consequence of that, the DW increases monotonously with current density. The velocity of the AF coupled DWs is enhanced as the saturation magnetization of the layers become similar in magnitude, and when their values decrease. Full realistic micromagnetic simulations indicate a faster coupled DW motion when the thickness of the spacer between the FM layers is reduced, and also when these layers exhibit equal saturation magnetization. While this later observation can be qualitatively described by the simple one-dimensional model, the first one is a direct consequence of the magnetostatic interaction between the internal magnetic moments of the DWs, which supports the antiparallel orientation between the internal magnetic moments in the AF coupling case. The conventional spin transfer torques does not significantly perturb the current-driven DW dynamics generated by the Slonczewski-like spin-orbit torque in AF coupled stacks, at least under perfect adiabatic conditions. It was also observed that inertia effects are significantly reduced in AF coupled stacks with respect to the single-FM-layer and FM coupling cases. The high efficiency of the current-driven DW dynamics in these AF systems is also coherent with the results obtained under in-plane longitudinal applied fields, which are also presented here.
Our micromagnetic simulations also indicate that up-down and down-up domain walls move with different velocities along a single-FM-layer stack with curved parts.
Moreover, domain wall tilting constitutes another important issue that interferes with the proper working of DW-based racetrack memories. This is particularly critical in the case of single-FM-layer stacks with curved parts, since this tilting may give rise to domain wall annihilation, and consequently, imposes a limit for the high density packing of domain walls.
Our micromagnetic simulations have also revealed antiferromagnetic coupling as a sound ally to avoid tilting and, consequently, to help the safe displacement of domain walls along such curved geometries. For these antiferromagnetic coupled stacks, up-down and down-up walls move with the same velocity along curved tracks at zero temperature. However, very close DWs can collapse even for AF coupling stacks under realistic conditions. The variation of the relative distance between adjacent walls is due to thermal fluctuations. Therefore, further systematic theoretical and experimental studies are needed to evaluate this limitation for strips with different widths and curvature radius.