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Stabilizing chiral spin-structures via an alternating Dzyaloshinskii-Moriya interaction

The stabilization of chiral magnetic spin-structures in thin films is often attributed to the interfacial Dzyaloshinskii-Moriya interaction (DMI). Very recently, however, it has been reported that the chirality induced by the DMI can be affected by dipolar interactions. These dipolar fields tend to form N\'eel caps, which entails the formation of a clockwise chirality at the top of the film and a counterclockwise chirality at the bottom. Here, we show that engineering an alternating DMI that changes sign across the film thickness, together with the tendency to form N\'eel caps, leads to an enhanced stability of chiral spin-structures. Micromagnetic simulations for skyrmions demonstrate that this can increase the effective DMI in a prototypical [Pt/Co/Ir] multilayer system by at least \SI{0.6}{mJ.m^{-2}}. These gains are comparable to what has been achieved using additive DMI, but more flexible as we are not limited to a select set of material combinations. We also present experimental results: by measuring equilibrium domain widths we quantify the effective DMI in [Pt/Co/Ir] multilayer systems typically used for skyrmion stabilization. Upon introducing an alternating DMI we demonstrate changes in the effective DMI that agree with our simulations. Our results provide a route towards enhancing the stability of chiral spin-structures that does not rely on enlarging the chiral interactions.

Magnetic skyrmions are whirling chiral spin-structures that can be as small as a few nm. [1][2][3][4] Because of their topological protection, they are extremely stable magnetic quasiparticles that might find their use in many applications such as magnetic racetrack memory. [1][2][3][4][5][6] Skyrmions are typically stabilized by the Dzyaloshinskii-Moriya interaction (DMI), which originates from a global inversion symmetry breaking in combination with spin-orbit coupling. [7,8] Although skyrmions exist in many systems, [2,[9][10][11][12][13][14] there is a great interest in skyrmions stabilized in ultra-thin ferromagnets. Their stabilization is achieved through the interfacial DMI from a symmetry-breaking interface between an ultra-thin ferromagnet and a heavy metal. [15][16][17] In these ultrathin systems, the magnetic properties can be tailored for specific applications by varying the magnetic layer thicknesses and interfaces.
Unfortunately, the DMI is often not large enough to stabilize magnetic skyrmions at room temperature. To compensate for this, the magnetic volume is usually increased to enhance the thermal stability and reduce the skyrmion energy. [12][13][14]18] The concomitant increase of dipolar interactions, however, has been shown both theoretically and experimentally to compete with the DMI leading to a non-uniform magnetic chirality across the thickness of the layers. [19][20][21][22][23][24][25][26][27][28] This is considered detrimental for applications because most of the functionality relies on the uniform chirality of a skyrmion across the thickness of the multilayer system. 1 On the other hand, despite their negative effect on the magnetic chirality, even without the DMI dipolar interactions are able to stabilize so-called dipolar skyrmions. [29][30][31] This occurs through the formation of Néel caps, which is the formation of a clockwise chirality at the top of the film and a counterclockwise chirality at the bottom. Inspired by this, we suggest here to combine the formation of Néel caps with a layer-dependent alternating DMI to enhance the stability of chiral spin-structures as is shown schematically in Fig. 1a for a magnetic domain wall. The dipolar fields introduce Néel caps with a clockwise (CW) Néel wall at the top of the film, and a counterclockwise (CCW) Néel wall at the bottom. For a uniform DMI this leads to a competition with the DMI across half the stack. Therefore, we intentionally reverse the sign of the DMI halfway through the system, such that in both halves of the stack the DMI field points in the same direction as the dipolar fields, which leads to a reduction in both the domain wall and skyrmion energy. In the first part of this Letter, we investigate this principle using MuMax 3 based micromagnetic simulations [32] and demonstrate that it leads to significant increases in the effective DMI by comparing skyrmion energies for different DMI configurations. Thereafter, we also present experimental results on the effect of modifying the DMI in a multilayer [Pt/Co/Ir] system. Upon changing the DMI configuration we find almost a factor 2 increase in the effective DMI after accounting for growth-induced variations in the magnetic parameters through an averaging approach, which we verify with micromagnetic simulations. This proves that Néel caps can be exploited to significantly increase the stability of chiral spin-structures and opens up a way to tailor them by modifying magnetic interactions on a layer-by-layer basis.
We investigate the behaviour of confined magnetic skyrmions in a 256 nm diameter circular dot (inset Fig. 1b) Fig. 1b the energy of the skyrmion state with respect to the uniformly magnetized state is plotted as a function of the DMI D for 3 different DMI configurations which are indicated in Fig. 1c: i) a uniform configuration, where the DMI is equal across all layers, ii) an enhanced DMI configuration, where the sign of the DMI aligns along the internal dipolar fields everywhere in the stack, and iii) a reduced DMI configuration where the sign of the DMI is always aligned anti-parallel to the dipolar fields. The enhanced and reduced DMI configurations lead to the formation of a thickness dependent chirality by the introduction of Néel caps. For all DMI configurations, the skyrmion energy decreases with increasing DMI as expected. In addition, the enhanced DMI configuration leads to a significantly reduced skyrmion energy, and the reduced configuration to an increase in the skyrmion energy. This is completely in line with the simple picture sketched in Fig. 1a. Specifically, as indicated in Fig. 1b with the DMI gain ∆D e , the D required to obtain a skyrmion whose energy is lower than the uniformly magnetized state decreases by 0.6 mJ m −2 upon introducing the enhanced configuration. This is a massive increase in stability and comparable in magnitude to the gains obtained when utilizing an effect like additive DMI. [13,18,33] Moreover, as we are not bound to the small set of material systems with a large additive DMI this should be more widely applicable. In supplementary note II we additionally show that while ∆D e  does vary with N , the presented behaviour remains qualitatively identical.
Introducing these different DMI configurations also has a profound effect on the skyrmion radius as shown in Fig. 2a. 2 In agreement with the results found in literature, an increase in D leads to an increase in the skyrmion radius. [13,34] We can understand this by thinking of a skyrmion as an out-of-plane (OOP) magnetized core enclosed by a domain wall. [34,35] As D increases the domain wall energy decreases, resulting in a skyrmion that can expand to enhance the dipolar coupling of the core to the annulus. The same mechanism explains the behaviour for the three different DMI configurations; the skyrmion becomes bigger (smaller) when introducing the enhanced (reduced) DMI configuration because the domain wall energy decreases (increases). In supplementary note IV we further illustrate that the confinement effect of the simulated dot does not qualitatively affect the presented behaviour.
We will now try to understand the effect of the different DMI configurations in a more general way. Combined, the behaviour depicted in Fig. 1b and Fig. 2a suggests some form of universality. All the curves show qualitatively the same behaviour as a function of D, apart from the shifted D values indicated by the arrows of DMI gain (∆D e ) and loss (∆D r ). This can be understood by considering the effect of the dipolar fields: as suggested by Lemesh et al., [25] the dipolar fields can be included as an effective DMI because both components introduce an effective in-plane magnetic field in the domain wall. We can thus introduce an effective DMI D eff = D + ∆D e,r (see Fig. 1b) for the enhanced and reduced configurations.
In this specific case, the enhanced configuration behaves as a system with an effective DMI D eff that is ∆D e ≈ +0.6 mJ m −2 larger than the DMI D because of the additive effects of the DMI and dipolar interactions. Conversely, the reduced configuration has a smaller effective DMI with ∆D r ≈ −0.2 mJ m −2 . A consequence of this universality is shown in Fig. 2b, where the skyrmion energy is plotted as a function of skyrmion radius for the three different DMI configurations. The simulations for the different DMI configurations collapse on the same curve. Thus, although for each configuration the D needed to obtain a particular energy/radius is different, the relationship between the two remains unaffected and can be described by an D eff that has a DMI-configuration dependent contribution. Last, in supplementary note V we demonstrate that the introduction of Néel caps can in some cases lead to non-circular skyrmions to accommodate both the DMI and dipolar interactions.
In the previous paragraphs we have introduced the unique ability of a layer-dependent DMI configuration to enhance the stability of skyrmions. This part presents experimental evidence which shows that D eff can be tailored by modifying the DMI on a layer-by-layer basis. To demonstrate this, instead of skyrmions we shift our attention to domain walls for their much easier experimental access, and their fully analogous underlying physics. They allow us to accurately quantify D eff because the domain width d in magnetic multilayers is determined by the competition between the domain wall energy and dipolar interactions between the domains. [12-14, 24, 25, 36-38] Here, we use the accurate stripe domain model by Lemesh et al. [38] to determine D eff in four different Ta(4)/Pt(2)/X/Ta(4) systems (thicknesses in parentheses in nm) to investigate the effect of the different DMI configurations. X for each stack is given by: Uniform II:  Table I  Indeed, D eff is larger for the enhanced configuration (1.5 mJ m −2 ) compared to the reduced configuration (0.9 mJ m −2 ). As suggested earlier, this possibly results from the modified DMI ordering in combination with the dipolar interactions that lead to Néel caps.
There is another element that we have to account for in this analysis: the effect of the stacking order of the layers. For both uniform configurations, |D eff | should be equal as we have simply inverted the stacking order. However, as we calculate |D eff | from the measured d we find that it varies by a factor of 2 for the uniform I and II configurations as is shown in Table I. Moreover, there is also significant variation in the measured saturation magnetization M s and anisotropy K for these configurations. From this we conclude that the magnetic parameters for [Pt/Co/Ir] and [Ir/Co/Pt] vary significantly because of growthrelated effects stemming from the stacking order. This seriously complicates the comparison between the different DMI configurations.
To still be able to compare the different DMI configurations, we need to extract the actual D values for the enhanced and reduced configurations to conclude if there is an increase in D eff for these systems. For this, we account for the growth-induced variations quantitatively by performing micromagnetic simulations in supplementary note VI. In these simulations we compare the domain wall energies in the enhanced and reduced stack with layer-dependent magnetic parameters, to a stack with the averaged parameters of both uniform stacks. From this we conclude that a hypothetical stack with these averaged magnetic parameters is a good approximation of the experimental situation, leading to errors in D eff < 0.1 mJ m −2 .
To calculate D for the enhanced and reduced configuration we can therefore simply average the D eff of both uniform configurations, which we have also included in Table I Summarizing the experimental part, we found changing the sign of the DMI halfway through a multilayer system is a powerful method to increase the effective DMI. With the relatively modest increase in D eff of the enhanced configuration (1.5 ± 0.1 mJ m −2 ) compared to the uniform I configuration (1.3 ± 0.1 mJ m −2 ) there is still room for improvement. The largest benefit of the enhanced DMI configuration will be found in a system with a large DMI where the growth-induced variations between the opposite stacking orders are small.
As there are a host of different interface combinations with a large DMI, we expect there to be many material combinations that fit this pattern.
We would now like to comment on two aspects of exploiting the Néel caps to stabilize chiral spin-structures. First, the introduction of an enhanced DMI configuration does not affect the skyrmion dynamics. Although the resulting vanishing total interfacial chirality suggests that spin-orbit torques can no longer be used to drive skyrmion dynamics, this is not true for the proposed experimental stacks of [Pt/Co/Ir] and [Ir/Co/Pt]. [24][25][26]43] When changing the stacking order halfway through the stack, it is not only the interfacial chirality, but also the local spin-orbit torques from the individual Pt and Ir layers that are reversed. [44] In this case, the spin-orbit torques acting on the skyrmion are the same for both halves of the stacks, ensuring skyrmions can still be driven efficiently using an electrical current. As there are also indications that an enhanced DMI configuration can postpone the Walker-breakdown-like behaviour for both domain walls and skyrmions to much higher current densities, [45] it is therefore interesting to explore their dynamics in the case of an enhanced DMI configuration in more detail.
Second, more ideas exist that make use of dipolar interactions and magnetic parameters that vary on a layer-by-layer basis. For example, by modifying the anisotropy, one might be able to increase the domain wall width at the top and bottom of the film to enhance the coupling with the dipolar fields and increase the skyrmion stability even further. Or, one could imagine changing the position within the stack where the DMI reverses, and thus the point at which the chirality reverses, to, for example, reduce the skyrmion Hall angle. [26] Perhaps it is even possible to stabilize more complex three-dimensional spin-structures such as the magnetic hopfion or skyrmion bobber by modifying individual magnetic parameters on a layer-by-layer basis. [46][47][48] In conclusion, using micromagnetic simulations we have shown that the stability of chiral Bloch and Néel configuration. 2 We minimized this state and compared the energy to the uniformly magnetized state.
The experimental systems investigated in the main paper, as well as the systems presented in Section S7 were DC sputter deposited using an Ar pressure of 2 × 10 −3 mbar on a Si substrate with a native oxide in a system with a base pressure of 4 × 10 −9 mbar. Pt and Ir were chosen because of their opposite interfacial DMI signs such that they favour CCW Néel walls for an Pt/Co/Ir stacking (+D) and CW Néel walls for a Ir/Co/Pt stacking (−D). 3-7 All samples were demagnetized using an oscillating decaying magnetic field applied at an 80 to 85°angle with respect to the film normal, starting from 5 T decreasing in 0.5 % increments with a cut-off at 0.5 mT. The resulting domain patterns for the individual samples were recorded using magnetic force microscopy (MFM) measurements under ambient conditions with custom-coated low-moment tips using a two-pass technique by recording the phase shift. 8 We extracted the periodicity using the Fourier method described in the supplementary of Ref. 9. The saturation magnetization M eff and anisotropy K were extracted from squid-VSM measurements using the area method. 10

S2. DIFFERENT REPEATS
In this section we show the results of simulations where we vary the number of repeats N .
The results for N = 2 system are plotted in Fig. S1a-c, for the N = 4 system in Fig. S1d-f, and the results for the N = 6 system are shown in Fig. S1g-i. Overall, the qualitative behaviour does not change with varying N . From Fig. S1a, d and g we find that the DMI values are reduced with increasing N due to increasing influence of the dipolar interactions; for that same reason we have bigger skyrmions for larger N (b, e and h). The curves for the different DMI configurations are still shifted with respect to each other, such that it is no surprise that the universal behaviour shown in Fig. 2b of the main paper is also valid for other N (Fig. S1c, f and i), which means that the interpretation of an effective D is more widely applicable.
Next, we investigate the dependence of the described DMI gain (∆D e ) and loss (∆D r ) on the number of repeats N . In Fig. S2  In this interpretation, the asymmetry between ∆D e for the enhanced configuration, and the stray fields of the domain walls themselves that prefers to align the walls in the Néel cap configuration. [11][12][13] For the reduced DMI configuration, this provides an energy gain offsetting the large energy loss of the other two contributions.

S3. FIELD DEPENDENCE
Here we illustrate that upon introducing the different DMI configurations there is also a significant effect on the field stability of a skyrmion. The skyrmion radius is plotted as a function of a magnetic field applied anti-parallel to the skyrmion core in Fig. S3. For all configurations the radius decreases with applied magnetic field to accommodate the Zeeman energy. Yet, for the enhanced configuration skyrmions are stable up to higher fields. Additionally, we find from these simulations that the smallest possible radius r c for a skyrmion does not vary noticeably with the DMI configuration, and remains constant at ≈ 4 nm for the investigated system. It must be noted, however, that at these length scales taken when interpreting these results below a radius of ∼ 10 nm. 1,14,15

S4. SMALLER DIAMETER DOT
In order to investigate the influence of geometric confinement on the behaviour discussed in the main paper, we present results for systems with a 128 nm diameter dot in this section.
For an N = 4 system the skyrmion energy is plotted as a function of D in Fig. S4a, and the skyrmion radius in Fig. S4b. Compared to the 256 nm dot of Fig. S1, we find that the stability region for skyrmions has shifted to larger D, and the skyrmion radius has decreased. These changes can be attributed to confinement effects within the dot which affect the skyrmions when their radii approach the dot radius. 16 Similar to the larger dots, the physical mechanism can once again be interpreted as leading to an effective DMI through the DMI gain (∆D e ) and loss (∆D r ). The universal behaviour of Fig. 2b of the main paper is therefore also present for smaller dots, as we show in Fig. S4c. Last, in Fig. S4d we set out the critical DMI values as a function of number of repeats N , which shows the same qualitative behaviour as Fig. S2. To conclude, although there is a quantitative difference due to the confinement effects within the dot, the qualitative behaviour as well as the benefits of an enhanced DMI configuration remain unaffected.

S5. NON-CIRCULAR SHAPE
We note a peculiar factor of the formation of Néel caps here that we believe has not been mentioned before in literature. When the effective fields from both the dipolar interaction and the DMI are approximately equal; i.e. when a Néel cap is about to be introduced, the shape of the skyrmion is no longer circular and varies across the different repeats of the multilayer system to accommodate both the DMI and dipolar interactions. We illustrate this in Fig. S5, where we find that the skyrmion shape is elliptical (in a circular dot) and that the profile in the top layer has rotated its long axis with respect to the profiles in the bottom layers. Although the effect is minimal, it is worth to keep in mind when using analytical models based on axial symmetric skyrmion shapes to explore their properties. 17  Table S1. (c) Zoomed version of (b), where we indicate the D shifts between the averaged and full stacks with D error .

S6. AVERAGING APPROACH VALIDATED
In this section we verify, using micromagnetic simulations, that the averaging approach advocated in main paper only leads to errors in the extracted D eff of at maximum,  We plot the energy for all these configurations in Fig. S6b  For the experimental D c = D exp = 1.0 mJ m −2 we find that the curves overlap very well, and we can quantify the error of the 'averaging' approach using the zoom-in shown in Fig. S6c. Here, we indicate these errors in the extracted D eff by D error , based on the difference in D c values between the full experimental stack (AB and BA) and the 'averaging approach' (C). We find in all cases that D error < 0.1 mJ m −2 , well within the experimental error bars. This means that the 'averaging' approach described in the main paper is fully justified.

S7. 8 REPEATS
In this section we demonstrate that the experimental increase and decrease in D eff observed in the main paper for N = 4 systems can also be found for N = 8 systems. To check this, we once again fabricated Ta(4)/Pt(2)/X/Ta(4) stacks, but with X for each stack now given by: In Fig. S7 we present the MFM scans, and in Table S2 the quantitative analysis following the same procedure as presented in the main paper. Similar to the N = 4 stack we find a large difference in the effective DMI between the enhanced and reduced stack with similar values of M s and K eff , completely in line with the predictions and the results of the N = 4 system. Furthermore, simulations suggest that variations in effective DMI for the N = 8 system should be slightly larger than the N = 4 system (not shown). The difference in effective DMI for the N = 8 system between the reduced and enhanced configuration is predicted to be approximately 1.0 mJ m −2 compared to 0.8 mJ m −2 for the N = 4 system. Experimentally, the variation for the N = 4 system of about 0.6 mJ m −2 agrees with the