Abstract
The topological Hall effect (THE) is the Hall response to an emergent magnetic field, a manifestation of the skyrmion Berryphase. As the magnitude of THE in magnetic multilayers is an open question, it is imperative to develop comprehensive understanding of skyrmions and other chiral textures, and their electrical fingerprint. Here, using Halltransport and magneticimaging in a technologically viable multilayer film, we show that topologicalHall resistivity scales with the isolatedskyrmion density over a wide range of temperature and magneticfield, confirming the impact of the skyrmion Berryphase on electronic transport. While we establish qualitative agreement between the topologicalHall resistivity and the topologicalcharge density, our quantitative analysis shows much larger topologicalHall resistivity than the prevailing theory predicts for the observed skyrmion density. Our results are fundamental for the skyrmionTHE in multilayers, where interfacial interactions, multiband transport and nonadiabatic effects play an important role, and for skyrmion applications relying on THE.
Introduction
Skyrmions are topologically protected, twodimensional (2D), localized hedgehogs and whorls of spin^{1}. Originally invented as a concept in field theory for nuclear interactions^{2}, skyrmions are central to a wide range of phenomena in condensed matter^{3,4,5}. Their realization at room temperature (RT) in magnetic multilayers^{6,7,8} has generated considerable interest, fueled by technological prospects and the access granted to fundamental questions. The interaction of skyrmions with charge carriers^{1,8,9,10,11,12} gives rise to exotic electrodynamics, such as the topological Hall effect (THE)^{13,14}. The topological protection of skyrmions results from the quantization of their topological charge (Q_{sk}), which counts the number of times the magnetization unitvector n(r) covers the unitsphere. Q_{sk} is determined by B_{eff}(r), the zcomponent of the emergent magnetic field that corresponds to the Berry phase accumulated by a spin tracking n(r)^{1,15,16}:
Here Φ_{0} = h/e is the flux quantum, h is Planck’s constant, −e is the electron charge. THE is the Hall response to B_{eff}(r). When charge carriers flow through a conductor with their spins tracking the skyrmion spin texture, the topologicalHall resistivity (ρ_{TH}) is^{1}:
where n_{T} is the 2D density of total topological charge. Here \(R_0{\!\!\prime} \) is an unknown Hall resistivity representing the effective density of charge contributing to THE. \(R_0^\prime \) is usually taken to be R_{0}^{13,17,18,19}, the ordinary Hall coefficient, which is extracted from the high field slope of the Hall resistivity (ρ_{yx}), and represents the total density of mobile charge. 0 < P < 1 is the spinpolarization of the charge carriers, and B_{eff}(r) is manifested through Φ_{0}. Assuming skyrmions are the sole carriers of topological charge Q_{sk} = 1 implies n_{T} = n_{sk}, the density of isolated skyrmions. Thus, within the adiabatic approximation, one expects a straightforward correlation between ρ_{TH} and n_{sk}. From the first observations in B20 systems^{13,14} to the recent multilayers, THE has been used as an indicator for the presence of skyrmions^{8,17}. However, a clear understanding of the effect is still lacking^{20}, especially in technologically viable multilayer films^{8,18,19}, where disorder and interface effects can play an important role^{7,21,22}. In particular, and as we explain below, the THE features in these multilayers are subtle and call for careful analysis of the transport measurements, as we demonstrate in Supplementary Note 3 and Supplementary Note 4. In contrast, the situation in some B20 systems is straightforward as they show a distinguishable characteristic hump in ρ_{yx} as a function of applied magnetic field (H) and temperature (T), corresponding to a skyrmion lattice^{23,24}. However, even in B20 systems such a feature does not appear always^{25}. One of our main goals here is to elucidate the significance of the subtle features that characterize our multilayer and to test their relationship to skyrmions.
Using magnetic force microscopy (MFM) and transport measurements, we present a comprehensive picture of the evolution of magnetic textures and their THE signature in a multilayer film capable of hosting skyrmions from RT down to at least 5 K. We demonstrate the relationship between n_{sk} and ρ_{TH} over a ≈200 K temperature range. As the applied field H is swept from saturation towards zero, we find that skyrmions aggregate in wormlike magnetic textures, which may carry large topological charge (Q_{W}), and manifest as peaks in ρ_{TH}. Quantitative modeling of these wormtextures uncovers qualitative agreement between ρ_{TH}(H, T) and n_{T}(H, T). Despite this, we find a large quantitative discrepancy indicating that the effect in multilayers is more involved.
Results
Multilayer system
Here we use sputtered [Ir(1)/Fe(0.5)/Co(0.5)/Pt(1)]_{20} (in parenthesis—thickness in nanometers; subscript denotes the number of repeats) multilayer films, with the composition chosen for exhibiting skyrmions across a large range of T. The RT characterization of the films indicates Dzyaloshinskii–Moriya interaction (DMI) D ≈ 2.0 mJ/m^{2}, and exchange interaction A ≈ 11 pJ/m^{8}. The effective magnetic anisotropy (K_{eff}) varies in the range ≈0.2–0.01 MJ/m^{3} as we change T from 5 K to 300 K (Supplementary Note 2). As demonstrated here, control over n_{sk} through variation of T is the key for unambiguous identification of the skyrmion THE signature. Without this control, the correspondence between n_{sk} and the subtle THE signal, which we establish by direct imaging, is impossible to uncover. In contrast to the B20 compounds, which host lattices of tubular Blochskyrmions^{26}, multilayers sustain skyrmions with tunable properties, and offer smoother integration with existing spintronic technologies. Spin textures in multilayers are influenced by interlayer dipolar and exchange interactions, magnetic frustration^{27}, and granularity^{7}, which can pin, stabilize, and deform the spin textures, and result in coupled pancakeskyrmions with different topologies^{22,27}. This complexity, and associated tunability, provide means for exploring the interplay between disorder, interactions, and topology.
Qualitative agreement between residual Hall signal and the magnetic textures
The magnetoresistance and Hall effect were measured using a lockin with nonperturbative current densities (≈10^{5} A/m^{2}). The presence of skyrmions is associated with an additional component in the measured ρ_{yx}^{13,14,17}. This contribution can be quantified by resolving ρ_{yx} into the ordinary (R_{0}H) and anomalous [R_{S}M(H)] Hall components^{13,17,18,19}, and ρ_{TH}:
We estimate ρ_{TH}(H) by Δρ_{yx}(H), the residual of the fit of ρ_{yx}(H) to \(\rho _{yx}^{fit}(H) = R_0H + R_{\mathrm{S}}M(H)\), which also yields R_{0} and R_{S} (Supplementary Figure 7). The accuracy of Δρ_{yx} is ensured by calibrating field offsets to avoid artifacts resulting from using different measurement setups (Supplementary Note 3 and Supplementary Figure 2). Our conservative estimate for the overall error in Δρ_{yx}, including a contribution from data analysis, is ±2 nΩ ⋅ cm, corresponding to the nonzero residual signal beyond saturation, where there are no skyrmions. Figure 1a shows Δρ_{yx}(H) at 5 K, with the overall features persisting to at least 300 K^{8}.
The features of Δρ_{yx}(H) in Fig. 1a are subtle. Here, we use MFM to determine the correspondence of these features with the magnetic texture. For this we followed the same field sweep from saturation as we did for Δρ_{yx}(H) and magnetization [M(H)]—the images were acquired at field increments as H was swept (Supplementary Note 1). Figure 1b–k shows the result for 5 K. Overall, we observe a similar evolution of the magnetic textures at T = 50, 100, 150, 200 K (Supplementary Figures 17 to 20).
We begin by comparing the magnetic textures to Δρ_{yx}(H). Beyond saturation, MFM shows the null signal expected for a polarized ferromagnet [Supplementary Figure 16(a)]. This is accompanied by the suppression of Δρ_{yx} (Fig. 1a) as expected for the topologically trivial polarized state. The onset of Δρ_{yx} commences at μ_{0}H ≈ −0.3 T with the nucleation of sub100 nm magnetic domains (Fig. 1b), which we identify as Néelskyrmions^{21}. By μ_{0}H ≈ −0.25 T (Fig. 1c) the increasing n_{sk} corresponds to a substantial Δρ_{yx} as expected from ρ_{TH} ∝ n_{sk}^{1}. Also, as n_{sk} increases, skyrmions aggregate to form wormlike features (Fig. 1c). By μ_{0}H ≈ −0.225 T, images show only worms (Fig. 1d, e). Surprisingly, at this field Δρ_{yx} peaks, suggesting a significant contribution from the worms which may have nontrivial topology. Meanwhile, the dense textures at intermediate H (Fig. 1f–h, and Supplementary Figures 17 and 18) correspond to reduced, yet finite, Δρ_{yx}. Careful inspection of such scans reveals wormlike features, to which we attribute the finite magnitude of Δρ_{yx} (Supplementary Note 7).
As H = 0 is approached, the worms evolve into labyrinthine helical stripes, and proliferate at the expense of the polarized background. This is coincident with the suppression of Δρ_{yx}, highlighting the close relationship between Δρ_{yx} and the magnetic texture. As H is increased towards positive saturation, the labyrinthine stripes evolve into worms, skyrmions, and eventually a uniformly polarized phase (Fig. 1i–k). For a texture with an opposite topological charge, the sign of Δρ_{yx} is reversed when H > 0. However, the MFM contrast does not change, due to the reversal of the tip magnetization near 0.1 T.
The distinct field ranges for isolated skyrmions (near saturation), worms (negative peak in Δρ_{yx}), and their coexistence (positive peak in Δρ_{yx}) (cf. Supplementary Note 6), offer a unique opportunity to compare the magnetic texture with Δρ_{yx}. In particular: (i) Does Δρ_{yx} track n_{sk}? (ii) How do worms produce such a large Δρ_{yx} ? (iii) Is there quantitative consistency between Δρ_{yx} and n_{T}?
To address (i) we exploit n_{sk}(T), which increases by an orderofmagnitude when we increase T from 5 to 200 K (Fig. 2). We attribute this proliferation to the suppression of K_{eff} with temperature^{8}. Interestingly, skyrmion size is only weakly Tdependent and does not change significantly within our resolution, not unlike theoretical predictions^{28}. However, further experiments with similar imaging conditions are required to confirm the size variation. Importantly, we find that Δρ_{yx}(T) tracks n_{sk}(T) over the entire range (Fig. 2a), and gives ≈0.6 nΩ ⋅ cm per skyrmion/μm^{2}. This correlation between skyrmion nucleation and the emergence of Δρ_{yx} strongly points towards the topological origin of the residual signal.
Magnetic worms and their topological charge
Having established the direct correspondence between n_{sk} and Δρ_{yx}, we now examine the worms. Though the presence of worms is expected in systems with competing interactions, such as in ferromagnetic films^{29}, their topological role is not obvious. In early MFM work on Bloch skyrmions in a B20 compound^{26}, helical stripe domains resulting from merging skyrmions were described by two halfskyrmions connected by a topologically trivial straight domain^{26}, and hence a topological charge of Q_{w} = ±1. This motivates us to examine whether worms carrying a topological charge given by Q_{sk} = ±1 can describe our results. We therefore plot n_{sk} + n_{w} (n_{w} is the number of worms per unit area) in Fig. 3a, with the sign chosen from the sign of Δρ_{yx}. As the plot shows, both n_{sk}(H) and n_{sk}(H) + n_{w}(H) do not track Δρ_{yx}(H). This calls for a closer look at the topological nature of the worms.
The following analysis of the topological charge of worms is motivated by sequences like Fig. 1b–d [also in Supplementary Figures 17(n)–(o)], which suggest that worms result from skyrmions clustering as n_{sk} increases. The transition of worms into typical stripe domains with Q_{w} = 1 requires a complete unwinding of their internal spin structure. The energy barrier for this suggests that the effective topological charge should be at least equal to the total number of skyrmions that form a worm (i.e., Q_{w} > 1). While skyrmions are expected to repel each other on very short lengthscales because of exchange coupling^{30}, clusters can be stabilized by attraction on an intermediate scale, due to exchange frustration^{31}.
Here our recent work, a magnetic multipole expansion of the field from skyrmions (MEFS)^{21}, provides a direct method to associate an effective Q_{w} with each worm. Figure 3b–g shows two typical examples where we fit the measured signal from worms by trains of skyrmions. Such images, which contain both skyrmions and worms, provide the foundation for this kind of analysis—the isolated skyrmions serve to constrain the fit amplitude per skyrmion, and improve the accuracy (Supplementary Note 7). Meanwhile, the analysis of images containing only worms (Fig. 1d) hinges on skyrmions–skyrmion repulsion on a length scale comparable to their radius^{30}. Therefore, the number of skyrmions clustered in a worm is determined by the total length of the worm, and the typical radius of skyrmions (≈40 nm^{21}). For images with densely packed features (e.g., Fig. 1f–h) identifying and extracting the worms themselves requires additional image processing, for which we employ a deeplearningmodelbased algorithm that extracts features relevant for classification from supplied examples (Supplementary Note 7). We note however, that while this analysis provides additional evidence for the correlation between magnetic textures and the Δρ_{yx} that we extract, our main conclusion does not rely on it. In all cases, our analysis allows us to compute the total n_{T}(H) from ∑Q_{w} + ∑Q_{sk}, where ∑Q_{w} is the topological charge of a worm after assigning an appropriate number of skyrmions to it, and we assume that all skyrmions (including those in worms) have Q_{sk} = ±1, with the sign chosen to match the sign of Δρ_{yx}.
To further verify our assertion of worms as trains of skyrmions, we have performed basic simulations of magnetization and the resulting evolution of magnetic textures in our multilayer stack. Simulations were performed using the MuMax^{3} package^{32} at T = 0 K for two different sets of magnetic parameters relevant to the magnetization of the stacks at different temperatures. In the simulations the field is swept incrementally towards saturation from H = 0. The magnetic texture in the simulations starts from labyrinthine stripes and evolves into worms, skyrmions and eventually a fully polarized state, in agreement with our experimental results. A complete set of simulated MFM images is shown in Supplementary Figures 10 and 11.
Figure 4 shows representative simulated maps of n_{z}(r), the corresponding MFM images, and the contribution to Q_{sk} from each individual domain. Dashed domains in Fig. 4c, f mark skyrmion trains which appear as worms in the MFM scans (Fig. 4b, e), clearly indicating Q_{W} > 1. Here, we point out that while these results are in qualitative agreement with the experimental evolution of magnetic textures and the formation of worms, the simulations do not account for disorder and thermal effects in the multilayers. Addition of such effects and a full theoretical treatment is necessary for a deeper understanding of skyrmion trains and skyrmion–skyrmion interactions.
The qualitative match between n_{T}(H) and Δρ_{yx}(H) (Fig. 3a), and the formation of worms from the clustering of skyrmions, as suggested by our micromagnetic simulations, reinforces our modeling of worms as trains of skyrmions. These results provide additional evidence that Δρ_{yx} results from skyrmion textures and is thus topological in nature, indicating Δρ_{yx} ≈ ρ_{TH}. Our observations on the emergence of worms with a potentially high topological number, previously not noticed experimentally, indicate they are distinct from trivial spin spirals, and form an essential part of the phase diagram for multilayer skyrmions^{33}. The presence of worms in a tunable multilayer offers a platform for studying skyrmion–skyrmion interactions over a wide parameter range, as well as applications, such as skyrmion racetracks^{7}.
Quantitative agreement between transport and imaging
Having explored the nontrivial topology of the worms and a qualitative match between ρ_{TH}(H, T) and n_{T}(H, T), we examine the quantitative match. As we show in Fig. 3a, the density of topological charge estimated from THE [Δρ_{yx}/(PR_{0}Φ_{0}), Eq. (2)] indicates a twoordersofmagnitude discrepancy. This implies that simply using Eq. (2) with \(R_0^\prime = R_0\) to understand the topological signatures of chiral magnetic textures in multilayer skyrmionhosts does not yield a comprehensive description of the measured Δρ_{yx}.
A possible culprit is the assignment \(R_0^\prime = R_0\) in Eq. (2)^{34}, which is justified for a single band material. This is not the case here: Bulk Fe and Co, the ferromagnetic ingredients of our multilayer stack, have several active electron and hole bands^{35}. In such materials R_{0} is suppressed because electrons and holes, which experience the same H, compensate each other’s contributions. The cancellation estimated from values reported for bulk Fe^{35} indicate suppression of R_{0} by an order of magnitude from the separate contributions of individual bands, partially addressing the discrepancy (Supplementary Note 8). Importantly, the cancellation may not happen in the same way for B_{eff}—the Berryphase can act differently on charge carriers from different bands^{23}, with an associated sensitivity to occupation^{36}. The fact that the peak value of Δρ_{yx}(H) changes by only ≈25% with T despite the sign change of R_{0}(T) (Fig. 2b), probably because of small variations of the occupations of the compensating bands, further confirms that \(R_0^\prime \ne R_0\). In this case, using R_{0} in Eq. (2) underestimates ρ_{TH}^{18,19}, although once \(R_0^\prime \) is determined this fundamental equation may still be used. For this it is essential to account for the electronic band structure of the chiral magnet.
Finally, we comment on the nature of spin transport in multilayers. Here, transport is influenced by various length scales such as the thickness of individual layers, their mean free path (l), charge and spin conductivities, spin diffusion length (l_{s}), and the size of the magnetic textures—including the domain walls and their chirality. In our case, the radius of the skyrmions is ≈40 nm and the domain wall width ≈7–14 nm \(\left( {\sqrt {A/K_{{\mathrm{eff}}}} } \right)\), where, A ≈ 11 pJ/m and ≈0.2–0.05 MJ/m^{3}. These length scales are comparable to the transport length scales reported for single thin layers (l_{Co} ≈ 5 nm and l_{Pt} ≈ 13 nm, l_{s−Pt} ≈1−10 nm)^{37}, indicating their relevance to the relationship between ρ_{TH} and n_{sk}. Spin scattering from interfaces and magnetic textures such as domain walls may cause spinflip scattering, resulting in a nonadiabatic, spin independent transport^{38,39,40}. As a result of the disordered skyrmion configurations, the charge carriers experience an inhomogeneous emergent field, in contrast to charge carriers in bulk systems which experience a systematically varying uniform emergent field due to the ordered arrangement of skyrmion textures. Notably, spin scattering in multilayer systems and its contribution to the anomalous Hall effect is dominated by skew scattering^{41}. Hence, a better understanding of this contribution may help resolve the discrepancy between the observed and the expected THE signal^{38,39,40,42}.
Discussion
Our complementary imaging and electrical transport studies provide clear evidence for the correlation between Δρ_{yx} and the topology of the magnetic texture in technologically viable magnetic multilayers. Furthermore, we elucidate the complexity of the Berryphase associated with the electrical fingerprint of chiral magnetic textures in those skyrmions hosting platforms. For a comprehensive understanding and in order to utilize THE emerging from magnetic skyrmions, it is imperative to consider (a) the band structure contributing to THE, (b) the possibility of Q_{sk} > 1^{27} and skyrmion–skyrmion interactions^{31}, (c) the coupling of skyrmions across layers and complex magnetic textures in buried interfaces^{22}, (d) the contribution from topologically trivial chiral configurations driven by magnetic spinfrustration^{20}, and (e) the validity of the commonly assumed adiabatic approximation^{20}.
Data availability
The authors declare that the data supporting the findings of this study are available within the paper, and its Supplementary Information.
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Acknowledgements
It is our pleasure to thank A. Petrović for inputs on transport experiments, as well as D. Arovas, A. Auerbach, ShiZeng Lin, D. Podolsky, M. Reznikov, and A. Turner for illuminating discussions. We also thank G. Goldman and Y. Schechner for help with image analysis. The work in Singapore was supported by the Ministry of Education (MoE)—Academic Research Fund (Ref. No. MOE2014T21050), the National Research Foundation—NRF Investigatorship (Reference No. NRFNRFI201504), and the A*STAR Pharos Fund (1527400026). Work at Technion was supported by the Israel Science Foundation (Grant No. 1897/14). We would also like to thank the Micro Nano Fabrication Unit, and to acknowledge support from the Russell Berrie Nano Technology Institute, both at the Technion.
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M.R., A.S., O.M.A. and C.P. designed and initiated the research. M.R. deposited the films and characterized them with A.S. and A.K.C.T. M.R. performed and analyzed magnetization and transport measurements with inputs from A.S., O.M.A. and C.P. A.Y. performed the low temperature MFM experiments with assistance from A.A. and A.Y. and O.M.A. analyzed the low temperature MFM data. F.M. performed micromagnetic simulations and A.Y. analyzed the simulated images. O.M.A. and C.P. coordinated the project. All authors discussed the results and provided inputs to the manuscript.
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Raju, M., Yagil, A., Soumyanarayanan, A. et al. The evolution of skyrmions in Ir/Fe/Co/Pt multilayers and their topological Hall signature. Nat Commun 10, 696 (2019). https://doi.org/10.1038/s41467018080419
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