Densely packed skyrmions stabilized at zero magnetic field by indirect exchange coupling in multilayers

Magnetic skyrmions in magnetic heterostructures are non-collinear chiral 2D-like topological spin textures that have attracted great attention in the last decade due to their remarkable properties, such as room-temperature (RT) stabilization, small size in the range a few tens of nanometers, current-driven mobility, and electrical detection.^1–4 Building on a rapid experimental progress, a variety of devices based on skyrmions have been conceptualized for encoding information, e.g., race track memories, logic devices, or neuromorphic computing.^5–10 In order to control the static and dynamical properties of magnetic skyrmions, different material systems have been investigated in order to stabilize different skyrmion configurations going from individual skyrmions to dense skyrmion lattices. Theses skyrmion systems can be ferromagnetic (FM),^3,11–15 ferrimagnetic,¹⁶ or more recently 2D materials.¹⁷ 

In most cases, the application of an external field of at least a few tens of mT is required for the stabilization of a skyrmion phase (SP) as the Zeeman energy is needed for the transition from the topologically trivial maze-domain configuration at zero-field. Nonetheless, in addition to the simple use of an external field, other approaches have been proposed for nucleation or stabilization of skyrmions such as current induced nucleation,^3,4,18 irradiation,¹⁹ probe interaction,^20,21 x-ray illumination,²² ultrafast laser pulses,²³ or artificial defects created by lithography,^24,25 among others.

From a practical point of view, the precise control of the external field allows us to finely tune the skyrmion size as well as the density.^15,26 On a more fundamental aspect, Büttner et al.²⁷ recently investigated the formation process of skyrmion lattices at pico-second time scale by combining an ultrashort laser pulse together with a static external magnetic field, which is required to break the time-reversal symmetry. Hence, beyond the fact that it may be an obstacle for the application of skyrmions in new types of computing devices, the use of an external field can also be complication to address experimentally some still-debated questions about the actual skyrmion nucleation mechanisms and their time scale.²⁸ To tackle this limitation, the stabilization of magnetic textures without any external field still remains an important challenge. Indeed, isolated skyrmions or skyrmionic bubbles in ferromagnetic multilayers (MML) have been already successfully stabilized at zero-field either in confined structures¹³ using the interlayer exchange interaction from a perpendicularly magnetized single films^29,30 or magnetic field history.^18,31–33 Note that for some applications, such as reservoir or neuromorphic computing fields,^34–36 there is a clear advantage of working with dense skyrmion ensembles instead of isolated skyrmions.

In this study, we demonstrate that zero external magnetic field (zero-field) SP can be stabilized at RT. To this aim, we first investigate the dependence of the skyrmion size and density on different parameters of the magnetic multilayers. The first objective is to reduce the field required for the SP stabilization down to a few tens of mT. Then, we describe the approach developed to stabilize zero-field SP in multilayers relying on an indirect exchange interaction generated by an additional bias layer (BL) with uniform perpendicular magnetization. The effective magnetic field created by the bias layer is electronically coupled to the skyrmion magnetic multilayers through a non-magnetic (NM) layer replacing the external field in stabilizing the SP. We demonstrate using magnetic force microscopy (MFM) the stabilization of zero-field SP with skyrmion diameter as small as 60 nm. Moreover, besides the small skyrmion size, we demonstrate that their density can be easily controlled by finely tuning the thicknesses of both ferromagnetic (FM) and NM layers, which is not the case for all the other approaches for having zero-field stabilization.

The magnetic multilayers have been grown by dc magnetron sputtering on thermally oxidized silicon wafers with 280 nm of SiO₂ under 0.25 mbar dynamic Ar pressure (the base pressure is 7 × 10⁻⁸ mbar). All the samples have a bottom buffer layer made of Ta(5)|Pt(8) and capped with 3 nm Pt layer to prevent oxidation, as schematized in Fig. 1(a). An alternating gradient field magnetometer (AGFM) and superconducting quantum interference device (SQUID) are used to measure the anisotropy field H_K and the spontaneous magnetization M_s. Magnetic imaging using MFM has been performed with low-moment magnetic tips in double pass tapping mode-lift mode. A custom made magnetic tip coated with a 7-nm thick CoFeB layer has been used for its low magnetic moment in order to limit the perturbation of the magnetic textures in the SP. The MFM setup is equipped with a variable external field module, which allows us to modify the external field on demand between two different measurements. The scanned area remains the same regardless small drifts due to the external field and small temperature variations.

The stabilization of the skyrmion configuration and its final characteristics is governed by the balance between the different magnetic energies, i.e., the direct Heisenberg exchange constant (A), the Dzyaloshinskii–Moriya interaction (DMI), the magnetic uniaxial anisotropy (K_u), and the dipolar energies, necessitating a precise magnetic characterization of the samples. The effective interfacial DMI (D_eff) as a function of the thickness (D_s = D_eff t_Co) has been measured by k-resolved BLS to be D_s = −1.27 pJ/m.^37,38 We notice that, even if the Pt thickness is only 0.6 nm, it results in an effective perpendicular anisotropy (PMA) and an amplitude of the DMI close to the value of thick layers (≃3 nm), as we previously demonstrated in Ref. 38. The thickness of Ru of 1.4 nm leads to a ferromagnetic Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two consecutive Co layers.³⁹ In recent work, we found that the exchange constant A depends on the FM thickness in the range that we consider here. This parameter is taken to be equal to A = 23 pJ/m from Ref. 38.

In order to characterize the magnetic properties of the MML as a function of t_Co, we grew a series of samples with n = 10 varying t_Co from 0.6 to 1.9 nm. A schematic view of the studied multilayers composed of a trilayer of (Pt(0.6)|Co(t_Co)|Ru(1.4)) repeated n times is shown in Fig. 1(a) (numbers in parentheses indicate the thickness of each layer in nm). Through the out-of-plane hysteresis loops shown in Fig. 1(b) together with in-plane hysteresis loops (Fig. S1), the important magnetic parameters can be determined, namely, the saturation magnetization (M_s), the out-of-plane (H_sat) and in-plane (H_K) saturation fields, the effective (K_eff) and uniaxial anisotropies (K_u), and the skyrmion density (ρ_sk) and diameter (D_sk) measured from the MFM image. Their evolutions with t_Co are shown in Figs. 1(c)–1(h). The saturation magnetization M_s t_Co [Fig. 1(c)] displays a quasi-linear increase with t_Co. Note that a null magnetization is extrapolated for finite t_Co ≈ 0.4 nm, suggesting that for this thickness, the Curie temperature (T_c) is below RT, similarly to what has been found for similar magnetic trilayers (see Ref. 40). The slope indicates the intrinsic M_s = 1.56 ± 0.01 MA/m, a value close to the bulk Co M_s value. The saturation field H_sat displayed in Fig. 1(d) with black open circles shows a continuous increment as a function of t_Co with a marked increase above 1.4 nm. The in-plane saturation field H_K represented with open blue circles decreases from 0.8 nm crossing H_sat at 1.6 nm, suggesting the spin reorientation transition (SRT) from out-of-plane to in-plane.

We then study the evolution of the effective anisotropy K_eff extracted from μ₀H_K = 2K_eff/M_s. This value is multiplied by Co thickness (K_eff t_Co) to be represented as a function of t_Co in Fig. 1(e). From t_Co = 0.6 to 1.0 nm, K_eff t_Co first increases, reaching the maximum value around t_Co = 0.9–1 nm. Thereafter, K_eff t_Co decreases linearly up to the largest Co thickness. We find the SRT from out-of-plane to in-plane (K_eff = 0) at t_Co = 1.53 nm in good agreement with other series of (Pt|Co|Ru)_×n that we studied recently.^38,39 The linear fit in Fig. 1(e) (blue dashed-dotted line) is expected to have a slope equal to $−μ0Ms2/2$ in the equation, considering the shape anisotropy and a purely interfacial PMA only. The estimated value is, in fact, not compatible with the value of M_s deduced from Fig. 1(c), indicating that an additional component is needed to explain the data in panel (e). Similarly, the calculated $Ku=Keff+μ0Ms2/2$ is not constant. Therefore, we need to consider the existence of a magneto-crystalline anisotropy constant K_u,c. The linear fit in Fig. 1(e) suggests K_u,c = −1.9 ± 0.1 MJ/m³ and K_u,s = 3.8 ± 0.1 mJ/m². The corrected K_u (K_uCorr) is represented with open squares in Fig. 1(f). The fitting dashed-dotted blue line corresponds to the expression K_u,s/t_Co + K_u,c. We analyze the evolution of the magnetic configuration as a function of t_Co under external field. In Figs. 1(i)–1(n) are shown the MFM images corresponding to t_Co = 0.6, 0.8, 1.4, 1.6, 1.8, and 1.9 nm, respectively. The objective is to determine t_Co allowing to turn the stripe domains into a densely packed skyrmion configuration at the intermediate state in the magnetization loop (⟨m_z⟩ = 0.5). For larger and moderate (but still positive) K_eff values, we find that only stripe domains or a combination of isolated skyrmions together with meander domains are stabilized, [see Figs. 1(i) and 1(j)] respectively. The SP can be stabilized at thicknesses smaller than the SRT with slightly positive K_eff (1.4 nm). However, large Zeeman energy is required to reach a SP, hence requiring field values near saturation. For ⟨m_z⟩ = 0.5, a moderate number of skyrmions combined with domains is visible in Fig. 1(k). Above SRT (i.e., K_eff < 0), the stabilization of SP at ⟨m_z⟩ = 0.5 is possible as shown in Fig. 1(l). The thicker the FM layer the higher the ordering and the density [Figs. 1(m) and 1(n)] for t_Co = 1.8 and 1.9 nm, respectively. The density is analyzed from Fig. 1(g), finding that it increases exponentially up to 40 ± 2 sky/μm⁻² in agreement with Ref. 41.

In order to determine the evolution of D_sk, we estimate their actual value from ρ_sk and the m_z value obtained in the hysteresis loop at the same field as the one used for the MFM images, i.e., $Dsk≈2(1−mz)/(πρsk)$ [open red circles in Fig. 1(h)]. Following this approach, the D_sk is found to decrease by 80% from t_Co = 1.4 to 1.9 nm. A second method to estimate the skyrmion diameter is to measure the full width at half maximum (FWHM) of the corresponding skyrmions from the MFM phase signal. The results are shown with red triangles in Fig. 1(h), presenting for both methods the same trend with a decrease in t_Co but less accentuated for the first one (20%). From this analysis, we select t_Co = 1.6 nm and n = 10 as it is the best compromise between order, density of the SP, and the external field requirements at ⟨m_z⟩ = 0.5.

Our objective is to optimize the stabilization of the SP and to minimize the required external field. To achieve this, we analyze the impact of the number of repetitions and the thickness of the lower Pt layer on the saturation field, the dipolar fields, and the anisotropy. By considering these factors, we can determine the effect on ρ_sk or D_sk of the SP. While increasing the number of repetitions increases thermal stability and signal-to-noise ratio, we find that it also implies an increase in the external field that will be needed to stabilize the skyrmions.⁴² As ⟨m_z⟩ = 0.5 is proportional to the out-of-plane saturation field, we study the variation of H_sat together with H_K extracted from the out and in-plane hysteresis loops in different multilayers with n = 1 to n = 20 repetitions (Fig. S2). As shown in Fig. 2(a), H_sat follows approximately a linear evolution up to n = 12 above which it remains constant. However, H_K shows a continuous linear increase without saturation, leading to an enhancement in both K_eff and K_u values [Fig. 2(b)]. For a number of repetitions larger than ten, K_eff becomes positive increasing from −0.009 to 0.5 MJ m⁻³. Similarly K_u increases from 0.36 to 5 MJ m⁻³. In Figs. 2(e)–2(h), we display the MFM images with n = 3, 5, 10, 15, and 20 as a function of the minimum applied external magnetic field μ₀H_z allowing the observation of the transition from labyrinth configuration into a SP, determining ρ_sk and D_sk. The result of the quantitative analysis is presented in Fig. 2(c), showing a monotonic increase (40%) in ρ_sk from n = 10 to 20. After studying the trend for the skyrmion diameter, it becomes apparent that the two methods exhibit contrasting trends. D_skMFM displays an upward trend with increasing n, which is consistent with Ref. 32. Both measurements intersect after n = 10, pointing to a dominant role of dipole field and in-plane field effects for samples with more than n = 10.

Following the results presented before, we will choose n = 3 for the rest of the study as being the best compromise between the impact of the interlayer dipolar fields and a good thermal stability. Thus, we can stabilize a dense SP with an external field that is four times smaller than for the sample with n = 20.

For this number of repetitions, H_sat can then be further reduced by decreasing the dipolar fields through the increase in the interlayer thickness of the NM layers. As shown Fig. 2(i), we find a 1/t_tot reduction of H_sat as function of the Pt thickness (t_Pt) ranging from 0.6 to 8 nm, t_tot being the total trilayer thickness. In this case, H_K remains almost constant. Therefore, for thicknesses greater than 1 nm, both K_eff and K_u remain constant [Fig. 2(i)] even if the Pt layer is increased up to 8 nm. (In Fig. S3 are shown the IP hysteresis loops confirming identical anisotropy values from t_Pt = 0.6–8 nm). In Figs. 2(m)–2(o), we present the MFM images showing a SP configuration for t_Pt = 3, 5, and 8 nm. Even though the reduction in the external field is about 60% between t_Pt = 3 and 8 nm, both ρ_sk and D_skMFM extracted from the MFM images are found not to change significantly [see Figs. 2(k) and 2(l)]. Note, however, that determining D_sk from m_z and ρ_sk, the diameter increases by more than 50% when the Pt spacer layer increases from 0.6 to 8 nm [see open red circles in Fig. 2(l)]. This suggests that the two methods differ more when the dipole fields are smaller i.e., for samples with lower number of repetitions. Then, we decide to use the following stacking sequence for the skyrmion MML: (Pt(8)|Co(1.6)|Ru(1.4))_×3.

Thanks to the previously described optimization of the MML properties, we find that a densely packed SP can be stabilized at relatively low external field values. We can thus envisage to replace such external field by a bias field generated by interlayer electronic coupling generated inside the MML by adding some uniformly magnetized layers. In Fig. 3(a), a schematic view of the complete sample allowing the zero-field stabilization of SP is presented. In addition to the already designed Pt|Co|Ru multilayered stack that hosts the skyrmions, it is composed of a bias layer (BL) grown on the buffer layer (Ta|Pt) and a NM spacer Pt coupling layer (CL) through which the indirect exchange coupling is modulated. We show here how the properties of the BL and CL may be engineered. We first optimize the BL aiming at reaching a strong enough effective field $(Heffbias)$ needed to stabilize the SP. The first important characteristic of the BL is that it should have a large PMA together with a completely uniform magnetization at remanence. The hysteresis loop of the BL composed of (Pt(0.4)|Co(0.6))_×4 is presented as the black open circle curve in Fig. 3(b). Note that we have chosen this final composition after having studied the BL properties as a function of the number of repetitions [see Fig. S4(a)] showing a squared shape with sharp transitions. The actual amplitude of $Heffbias$ has been experimentally estimated following the procedure that we developed in Ref. 39. Further information can be found in Figs. S4(c) and S4(d) of the supplementary material. The next step is to optimize the thickness of the CL through which the BL is electronically coupled to the skyrmion MML. We know from the above discussion that the H_eff amplitude required to stabilize the SP is ≈25 mT. The CL thickness determines directly the amplitude of the effective bias field $Heffbias$ acting on the bottom of the skyrmion MML. The evolution of $Heffbias$ as a function of the $tPtbias$ coupling layer is presented in Fig. 3(c) in which we see that $Heffbias$ decays exponentially, almost vanishing at $tPtbias=3$ nm. Note that the choice of the CL thickness influences indeed also the magnetization reversal process of the complete MML system. For example, we find that for $tPtbias<2.2$ nm, the effective field $Heffbias$ is very large, making that the BL and the skyrmion multilayer are sufficiently coupled so that their magnetizations reverse simultaneously; hence, no skyrmion can be stabilized. In the inset of Fig. 3(c), we display the experimental results showing $Heffbias$ vs $Hzswitch$⁠. We define $Hzswitch$ as the external field applied when the BL magnetization switching reversal occurs. The interesting coupling regime is when the BL and the skyrmion MML switch independently, keeping a large enough bias field. It corresponds to $tPtbias$ ranging from 2.2 to 3.0 nm. More details about the reversal mechanisms can be found in Fig. S4(b), where the loops are labeled with arrows pointing at $Hzswitch$⁠. The hysteresis loop of the complete system with a CL of $tPtbias=2.3$ nm is displayed with red circles in Fig. 3(b), where the blue circle curve is the loop of the skyrmion MML. The experimental procedure to prepare magnetically the system leading to the zero-field stabilization of SP consists first in a complete saturation of the MML magnetization followed by sweeping the external field back to zero [red arrows in Fig. 3(b)]. Red circles are initial and final magnetization state.

In Fig. 3 are presented the MFM images of the as-grown magnetization state (d)–(f) and at remanence after saturation (g)–(i) for three CL thickness, $tPtbias$ = 3.0, 2.5 and 2.3 nm. First, we see that in the as-grown state, some large domains (⁠$≈μ$m size) are present in the BL coexisting with smaller labyrinthine domains from the skyrmion MML. On the contrary, after having saturated the system applying H_z ≥ 80 mT and returning to remanence [see Figs. 3(g)–3(i)], the bias layer magnetization remains monodomain and only the magnetic configuration from the skyrmion MML is detected by MFM. For $tPtbias=3.0$ nm, the remnant structure consists of a maze-domain configuration [see Fig. 3(g)]. For $tPtbias$ = 2.5 nm, the magnetic configuration is mainly composed of skyrmions together with elongated domains [see Fig. 3(h)]. Finally, for $tPtbias=2.3$ nm presented in Fig. 3(i), the MFM image clearly indicates that the zero-field SP is stabilized.

In this part, we aim at investigating how ρ_sk and D_sk can be adjusted by finely tuning various MML parameters. Beginning with the optimized system described above, i.e., BL|Pt(2.3)|[Pt(8)|Co(1.6)|Ru(1.4)]_×3, we can slightly modify anisotropies and dipolar fields in two ways: (i) by increasing t_Co using the system BL|Pt(2.3)| [Co(1.7)|Ru(1.4)|Pt(8)]_×3 and (ii) by reducing the distance of the FM layers varying the NM bottom layer thickness of the trilayer from 8 to 5 nm. This yields the experimental system BL|Pt(2.3)|[Co(1.6)|Ru(1.4)|Pt(5)]_×3. We first analyze the zero-field SP’s statistics of the optimized sample. In Fig. 4(a), we present the corresponding MFM image, while the distribution of the number of skyrmions as a function of D_skMFM is shown in Fig. 4(b). The diameter distribution can be accurately fitted with a Gaussian function, resulting in a mean diameter of 85 ± 5 nm. Based on the analysis of a 5 × 5 μm² MFM image, the density is found to be 10 μm⁻², slightly higher but in agreement with the same system without the BL. In Fig. 4(c), we display the resulting SP imaged by MFM at zero-field after saturation for the sample with t_Co = 1.7 nm. As expected, the correspondingly modified anisotropies and interlayer dipolar fields lead to different characteristics of the zero-field SP. The D_sk,MFM distribution in Fig. 4(d) can be fitted with a Gaussian function with mean diameter $≈90±5$ nm and with a density of 7.0 μm⁻². Note that there is a reduction of about 30% of ρ_sk by increasing t_Co 0.1 nm (6%). Finally, in Fig. 4(e) is shown the MFM image of the system with t_Pt = 5 nm. Note that for this sample, we had to use a lower magnetization tip to measure without no apparent disruption of the magnetic configuration, leading to a much lower MFM contrast. In this case, the D_skMFM distribution is fitted with an asymmetric Lorentz 2σ function presenting a mean diameter 60 ± 5 nm and a density of 12 μm⁻², leading to a reduction in the apparent MFM radius of almost 30%. Note that using 5 nm thickness of the NM bottom layer, the required BL $Heffbias$ is larger than the one for 8 nm [Figs. 2(n) and 2(o)]; hence, there are a few remaining wormy-like domains.

Finally, we discuss the role of the BL on the skyrmion characteristics. ρ_sk and D_sk are related to κ, that is, the critical parameter describing the skyrmion stability $(κ=πD/4AKeff)$⁠.^15,43,44 Using the values that have been determined experimentally, we calculate that the κ parameter ranges between 0.2 and 0.35 at the maximum. These relatively low values seem to indicate that the SP should be considered as a configuration with a large density of isolated skyrmions rather than a real skyrmion lattice phase, which is expected only for κ larger than 1. Note, however, that if K_eff is used in the κ calculations instead of K_u, as performed often, such estimation of κ cannot be performed in case K_eff < 0. The behavior of ρ_sk and D_sk with respect to K_u is represented in Figs. 4(g) and 4(h), respectively. The general trends for ρ_sk and D_sk following a negative slope are in good agreement with results found by other groups.^41,42 However, all the points associated with BL+MML (open symbols) are shifted toward larger values, being K_u more than double. This indicates a potential increase in the effective anisotropy because of the coupling with the BL. The optimized BL+MML sample (open red circles) has a similar density than the MML (red squares), however, showing a smaller D_skMFM. The sample with t_Co = 1.7 nm (yellow) presents a different behavior with and without BL. The MML is fully in-plane (K_eff < 0), and having only three repetitions, few skyrmions are stabilized. However, in the presence of the BL, the skyrmion density is double. Another interesting result is that D_skMFM does not decrease compared to the optimized sample. This could indicate that the anisotropy induced by BL plays an important role in the minimum energetic state of the skyrmions at remanence. Similar results are observed for samples with t_Pt = 5 nm (purple); the resulting ρ_sk follows MML behavior; however, it is the D_skMFM where the difference is more accentuated with a large decrease. This fact together with the asymmetric fit in the distribution of radii indicates a large variation in the remnant energetic state due to the BL.

In conclusion, we thoroughly explore how to stabilize a densely packed skyrmion phase in a MML without requiring an external magnetic field. We describe how by adjusting the properties of the magnetic layers in the MML (Pt(t_Pt)|Co(t_Co)|Ru1.4)_×n with a moderate number of repetitions, specifically n = 3 and a larger t_Pt = 8 nm, we are able to precisely control the size and density of the skyrmion phase at zero-field. This is achieved by generating an effective magnetic field via an electronically coupled, uniformly magnetized bias layer, which effectively transforms the labyrinthine domain configuration into a dense skyrmion phase configuration. By slightly varying the thickness of the trilayer, the skyrmion diameter and density can be tuned, leading to a variation of $Heffdip$⁠. Such precise control of ρ_sk and D_sk at zero-field is an important outcome of this work and could, for example, facilitate the investigation of fundamental mechanisms and the time scale required to overcome the topological barrier leading to the nucleation of a skyrmion phase. The presented approach might also present some compelling opportunities for neuromorphic computing applications based on skyrmion phase-based systems.