Abstract
Magnetic skyrmions are topological quasiparticles of great interest for data storage applications because of their small size, high stability, and ease of manipulation via electric current. However, although models exist for some limiting cases, there is no universal theory capable of accurately describing the structure and energetics of all skyrmions. The main barrier is the complexity of nonlocal stray field interactions, which are usually included through crude approximations. Here we present an accurate analytical framework to treat isolated skyrmions in any material, assuming only a circularlysymmetric 360° domain wall profile and a homogeneous magnetization profile in the outofplane direction. We establish the first rigorous criteria to distinguish stray field from DMI skyrmions, resolving a major dispute in the community. We discover new phases, such as bistability, a phenomenon unknown in magnetism so far. We predict materials for sub10 nm zero field room temperature stable skyrmions suitable for applications. Finally, we derive analytical equations to describe currentdriven dynamics, find a topological damping, and show how to engineer materials in which compact skyrmions can be driven at velocities >1000 m/s.
Introduction
Magnetic skyrmions are spin configurations with spherical topology^{1,2,3,4,5}, typically manifesting as circular domains with defectfree domain walls (DWs) in systems with otherwise uniform outofplane magnetization. Skyrmions are the smallest nontrivial structures in magnetism and they behave like particles^{6,7,8,9,10,11}, which makes them of fundamental interest and of practical utility for highdensity data storage applications^{12,13,14,15}. Skyrmions have been investigated for decades^{16}, but only recently has attention shifted to the detailed domain wall structure and therebydetermined topology. Two factors have driven this trend: technological advances allowing for direct imaging of the spin structure^{17,18,19} and the discovery that the DzyaloshinskiiMoriya interaction (DMI) can be used to stabilize that structure. In particular, DMI can lead to a skyrmion global ground state above the Curie temperature (T_{ c }) in a GinzburgLandau theory of a ferromagnet^{20,21}. Stray fields are not included in that model, but are of critical importance for understanding skyrmions in real materials, as underlined by the fact that all isolated room temperature skyrmions so far were observed in relatively thick films with sizable saturation magnetization^{11,22,23,24,25,26,27,28} and hence very strong stray field interactions. To explore the full skyrmion phase diagram and understand skyrmion stability, a theory with accurate predictive power is required and such a theory must include stray field interactions.
Stray field energies are the most difficult to treat analytically due to their nonlocal nature: they involve sixdimensional integrals whose kernel locally diverges. All existing models involve approximations that are motivated only by the need to simplify, without considering the error of the approximation or the limits of validity^{16,20,29,30,31,32,33,34,35,36,37}. Tu^{38}, Kiselev et al.^{39}, and Guslienko^{31} evaluated the full stray field integrals, but only numerically, which is prohibitively slow and technically demanding. Micromagnetic simulations are hence the only generally applicable tool to obtain quantitative predictions, but they are too slow to systematically examine skyrmion properties across a parameter space comprising four material parameters, film thickness, and magnetic field.
Here, we derive a fully analytical theory with the precision of micromagnetic simulations but orders of magnitude faster performance, providing unique access to the full skyrmion phase diagram and deep insights into the underlying physics. For instance, we can mathematically prove that there are two types of skyrmions, stray field skyrmions and DMI skyrmions, and we discuss how to experimentally distinguish between them. Our theory predicts a sharp transition separating these two skyrmion phases, and a phase pocket in which they can coexist, leading to bistabilities and zerostiffness skyrmions. These new states exist at room temperature and can have many novel applications, some of which we suggest here. In view of applications, we prove that the Cobased multilayers at the focus of most current experimental efforts are incapable of hosting roomtemperaturestable sub10nm skyrmions. However, by examining >10^{6} material parameter combinations, we identify alternative materials suitable to host such skyrmions at room temperature without a stabilizing field. Finally, we derive simple analytical equations to describe currentdriven skyrmion dynamics with accuracy comparable to micromagnetics but yielding key insights that cannot be gained from numerics. We discover a topological contribution to damping that severely reduces the mobility of small skyrmions, and propose materials that can mitigate this effect to permit sub10 nm skyrmions to be driven by current at >1 km/s with vanishing skyrmion Hall angle.
Results
Validation of the model
Our theory predicts the energy of isolated skyrmions in a film of arbitrary thickness and infinite inplane extent, based on the recent experimental confirmation^{19,23} of an analytic and universal 360° domain wall (DW) model^{40} for the spin structure of all skyrmions. Although the 360° DW model is not an exact solution of the micromagnetic energy functional, it shows excellent agreement with experiments^{19,23} and is validated by our extensive micromagnetic simulations. Taking this model as an ansatz for the skyrmion spin structure, we derived analytic expressions for the total energy function with better than 1% accuracy over the entire parameter space (see Supplementary Information), assuming only that the structure does not vary along the outofplane direction. For a given set of material parameters (uniaxial anisotropy constant K_{ u }, saturation magnetization M_{ s }, exchange constant A, interface and bulk DMI strengths D_{ i } and D_{ b }, and magnetic layer thickness d) and external field H_{ z }, minimization of this function yields the equilibrium skyrmion configuration. Multilayers are included straightforwardly via the effective medium approach^{24,41} (for the limitations of the effective medium model, see ref.^{41}). Our model is therefore generally applicable to skyrmions in any material, with the exception of thick multilayers with very strong stray fields and weak DMI where flux closure surface states form and where the assumption of uniform magnetization along the outofplane direction does not hold^{42}. In the Supplementary Information, we provide analytical expressions for both bulk and interface DMI terms; in the discussions below, we apply our model with a focus on systems with interface DMI only, but the model itself is more general.
The spin structure is parameterized by its radius R, domain wall width Δ, domain wall angle ψ, and topological charge N (Fig. 1a). R and Δ determine the magnetization profile m_{ z }(x, y), whereas ψ specifies whether the inplane component of the domain wall spins is radial (Néel, ψ = 0, π), azimuthal (Bloch, ψ = π/2, 3π/2), or intermediate. Although the original 360° domain wall model included an analytical function Δ(R)^{40}, we find that stray field interactions lead to a highlynontrivial dependence of Δ on R and they must be treated independently. For large ρ = R/Δ, skyrmions consist of an extended outofplane magnetized domain bounded by a narrow circular domain wall, while for \(\rho \sim 1\) the inner domain is reduced to a pointlike core. We refer to these limiting cases as bubble skyrmions and compact skyrmions^{43}, respectively, consistent with the literature, but note that many skyrmions observed recently^{11,22,24,44} showed intermediate values of ρ and cannot be classified distinctly.
Figure 1b shows that our model agrees accurately with micromagnetic simulations and the experimental data of Romming et al.^{19}. Our model yields the energy of a given skyrmion configuration in less than a millisecond on a regular personal computer and the full energy landscape in a few seconds, providing dramatic improvement over micromagnetic simulations. Moreover, it provides information not easily accessible by simulations. For example, since Δ and ψ can be minimized for any nonequilibrium R, one readily obtains the function E(R) that describes skyrmion stability and rigidity. By contrast, standard micromagnetic simulations only yield the equilibrium spin structure.
Figure 1c shows E(R) for the skyrmion in Fig. 1b at μ_{0}H_{ z } = −2 T, giving insight into skyrmion stability. The individual energy contributions are plotted in Fig. 1d. There are two stable states: the skyrmion state at R_{eq} and the ferromagnetic ground state at R = 0. Despite their different topology, there is a path from one to the other through the singular R = 0 state. This singular state lacks a topology, hence topological quantization is lifted here. As seen in Fig. 1d, all energy terms vanish as R → 0 except for the exchange energy, leading to a finite, universal skyrmion energy at R = 0,
This value is close to the absolute minimum 8πAd of the exchange energy of a continuous spin structure with integer topological charge^{3,36}, and much smaller than the 38Ad predicted using a simple linear skyrmion profile^{33}. Our identification of a topologically valid and energetically possible path to annihilation clarifies that all skyrmions can be annihilated, even in continuum models, despite the common perception that topological protection guarantees stability^{32,45,46}. In particular, although E_{0} relates to topology as previously discussed by Belavin and Polyakov^{3}, it corresponds to the skyrmion nucleation energy barrier E_{ n }. Skyrmion stability, however, is related to the annihilation energy barrier E_{ a } = E_{0} − E(R), which depends nontrivially on all micromagnetic parameters and has no direct topological origin.
The annihilation barrier in Fig. 1c corresponds to only \( \sim 3{k}_{B}{T}_{{\rm{RT}}}\) at T_{RT} = 300 K, and hence the small skyrmions observed by Romming et al.^{19} can only exist at cryogenic temperatures. We focus in the remainder of this work on roomtemperature stable skyrmions, i.e., those with E_{ a } > 50k_{ B }T_{RT} corresponding to lifetimes >10 years, as required for applications. We note that E_{ n } and E_{ a } serve as upper bounds for the true energy barriers because skyrmions can deform in a way that is not covered by the 360° domain wall model, hence reducing the nominal energy barrier^{47}. However, previous studies^{47} and our own micromagnetic simulations indicate that deformations reduce the energy barrier by <2Ad. Note that the collapse energy E_{0} in the atomistic simulations by Rohart et al.^{47} is 23Ad and 22Ad with and without deformations, respectively, significantly smaller than the continuum limit of 8πAd, which the authors attribute to lattice effects. Such effects are beyond the scope of the present continuum theory, and therefore, we limit our subsequent discussion to the validity range of the micromagnetic framework, i.e., to diameters of approximately 1 nm and larger, and subtract 2Ad from all energy barriers in an attempt to include possible deformations.
Smallest roomtemperature stable size – unifying existing models
Figure 2 shows that our model reproduces the limiting cases treated previously in the literature and continuously connects them to provide otherwise inaccessible insights into the entire skyrmion phase diagram. Two common approximations exist to describe circular spin textures in outofplane magnetized films. The first is the wall energy model^{48} introduced in the 1970s to treat bubble domains. This model approximates the domain walls as infinitely thin with constant energy per unit length σ_{DW}, enabling exact solutions for the stray field energies^{48}. A characteristic feature is that E(R) starts from zero with positive slope at R = 0 and goes through a maximum before reaching a minimum at R = R_{eq}. This minimum vanishes above a critical field and the collapse diameter is finite even at T = 0. The second model, derived by Bogdanov and Hubert^{34,35} for small skyrmions in ultrathin films, approximates stray field energies as effective anisotropies. The original model does not include analytic solutions, but we derive these equations in the Supplementary Information. The effective anisotropy model predicts isolated skyrmions for any finite DMI^{34}, and they cannot be annihilated by applied fields at T = 0.
The room temperature collapse diameter as a function of D_{ i } and film thickness is shown in Figs 2a,b for the wallenergy model and effective anisotropy model, respectively, using the most accurate available form of the former^{41}. These figures correspond to a 50k_{ B }T_{RT} stability criterion; a 30k_{ B }T_{RT} criterion (lifetime of order seconds) leads to similar results. The wall energy model predicts roomtemperature stable skyrmions for all D_{ i } and d, with a variable but sizeindependent ψ. These skyrmions are always relatively large. The effective anisotropy model, by contrast, predicts roomtemperature stable skyrmions with size <10 nm, but only above a critical D_{ i } that is quite large in ultrathin films. With interfacial DMI, skyrmions in this model are always purely Néel (see Supplementary Information).
The predictions for our theory are shown in Fig. 2c. The most obvious feature is a sharp boundary separating \(\ll 10\,{\rm{nm}}\) and \(\gg 10\,{\rm{n}}{\rm{m}}\) skyrmions. This phase boundary qualitatively follows the stability boundary in Fig. 2b, and the predicted collapse diameter above and below this contour agrees surprisingly well with the effective anisotropy and the wall energy model, respectively. It would therefore be tempting to use Bogdanov and Hubert’s model whenever it predicts room temperature stable skyrmions and the wall energy model in all other cases. This hybrid approximation would predict a collapse diameter diagram similar to the correct result in Fig. 2c, but there are important properties that this approach would omit. First, the transition from one model to the other is not always as sharp as the collapsediameter figure suggests. At fields below the collapse field, large and small skyrmions can coexist, yielding the bistable region denoted in Fig. 2c. A hybrid model would be intrinsically incapable of describing this phase since it can only describe one type of skyrmion at a time. Second, the domain wall width and domain wall angle can strongly depend on the skyrmion radius, which is not predicted by either model.
Figure 2d shows how E(R) evolves with increasing DMI and how mixed states appear. Without DMI, E(R) starts with zero or positive slope at R = 0, then reaches a maximum and diverges logarithmically to −∞ if H_{ z } = 0. An applied field can compensate the divergent stray fields and stabilize a minimum at large R (not shown in the figure). With DMI, the situation changes qualitatively: the slope at R = 0 becomes negative so the minimum appears before the maximum. Initially, the minimum is too shallow and appears in condition of unstable Δ (see Supplementary Information section 6). This is the case for D_{ i } = 1.2 mJ/m^{2} in Fig. 2d. For larger D_{ i }, the minimum becomes deeper and shifts to larger R. At first, this new minimum is stable only at low fields (D_{ i } = 1.5 mJ/m^{2} in Fig. 2d) but collapses before the stray field minimum at large R vanishes. That is, the smallest skyrmions are not necessarily found at the largest possible field values. At even larger D_{ i }, the minimum at small R becomes more pronounced and can become energetically degenerate with a second minimum, as shown for D_{ i } = 2.5 mJ/m^{2} and D_{ i } = 3.0 mJ/m^{2}. The important difference between these two states is that the energy barrier separating the minima is much larger than 50k_{ B }T_{RT} for the lower DMI value (D_{ i } = 2.5 mJ/m^{2}) and almost vanishingly small for the larger D_{ i }. Those states constitute a bistability and a zero stiffness skyrmion, respectively, as discussed in more detail below. At even larger DMI, the maximum between the two minima vanishes and only one stable state remains.
The spin textures in Fig. 2c are all topologically equivalent: they exhibit spherical topology and unity topological charge and hence are all magnetic skyrmions. However, there has been much disagreement in the literature as to whether different stabilization mechanisms lead to fundamentally different types of skyrmions, and, if so, how to distinguish between them. Kiselev et al.^{39,49} suggested to use the skyrmion profile to determine its type. Similar to Guslienko’s study on stray field skyrmions in magnetic dots^{31}, we find that the radial profile is not a robust distinguishing criterion, as shown in Fig. 2e. We find that stray fields can stabilize compact skyrmions in the absence of DMI, with sizes down to ~20 nm, despite frequent statements to the contrary^{22,39,50}. On the other hand, stray field skyrmions with DMI below the transition line in Fig. 2c can have purely Néel domain walls, and possess either a compact or bubblelike profile (Fig. 2e). The profile can be continuously tuned from one to the other without changing the topology, either by applied field or by adjusting \(\kappa =\frac{\pi {D}_{i}}{4\sqrt{A{K}_{u}}}\) (Fig. 2e, inset), as also shown in^{34,35}. Hence, skyrmions with DMIstabilized Néel domain walls are not necessarily DMIstabilized skyrmions, even if their profiles are compact, since the energy giving rise to the minimum in E(R) can still derive from stray fields.
Based on Fig. 2c and the underlying energetics, we offer here mathematicallyprecise and experimentallyaccessible criterion to categorize skyrmions unambiguously, defining the terms “DMI skyrmions”, “stray field skyrmions”, and “bubble skyrmions” as labeled in Fig. 2. The mathematical definition is based on the understanding of Fig. 2d. We first note that all contributions to the total energy can be classified into domain wall energies (\(\propto \,R\) at large R) and bulk energies (all other energies), see also Figs 1c and d. Exchange, anisotropy, DMI, and volume stray field energies are domain wall energies. The Zeeman energy is a bulk energy. Surface stray fields contribute to both categories: the domain wall contribution leads to an effective reduction of the anisotropy and the bulk contribution effectively reduces the external field. We see that the minimum at small R is formed by the local energies while the minimum at large R is essentially a minimum in the bulk energies (which can be shifted due to the linear slope of the domain wall energies), see Fig. 2d. We therefore identify DMI skyrmions as minima of the domain wall energy. Stray fields may still play an important role in shaping this minimum and giving it the required depth for thermal stability, but the origin is still the minimum in the domain wall energy term that can only appear due to DMI. We include in this definition skyrmions whose domain wall energy has a negative slope at the minimum. By this definition, we include all skyrmions in materials with globally negative domain wall energy density. Physically, these states behave like minima in the domain wall energy, but they require a magnetic field to stabilize isolated skyrmions against expanding to helical states or skyrmion lattices. Finally, we call an energy minimum a “mixed state” if it is formed by minima of both domain wall and bulk energies (e.g., when both minima are at the same radius or when the energy barrier in between is so small (compared to k_{ B }T) that they form one extended stable state).
The domain wall energy minimum is the key aspect that the wall energy model neglects in its approximations. Its origin lies in “domain wall compression” wherein the reduction of Δ at small R leads to the zero radius energy of 27Ad (instead of 0 in the wall energy model). Domain wall compression leads to a steep increase of the domain wall energy (mostly exchange) at small R, which is why domain wall energy minima can be thermally stable down to much smaller radii than minima in bulk energies. This is illustrated in Fig. 1c. Even when considering that the ground state skyrmion energy already includes some compression energy, the difference between a linear interpolation of the domain wall energy according to the wall energy model and the correct result is 17Ad (27Ad instead of 10Ad). This additional 17Ad in stabilizing energy is one of the reasons why DMI skyrmions are physically different from stray field skyrmions.
Our mathematical definition is directly linked to discernibly different physical properties. First, the radius and in particular the collapse radii are different. Second, the collapse field shows distinct scaling: it is almost independent of DMI for stray field skyrmions and strongly increases with DMI for DMI skyrmions (see Fig. 3a). Third, Δ is determined by a competition between exchange and anisotropy for stray field skyrmions, but for DMI skyrmions it reflects a balance between DMI, anisotropy, and external field. That is, Δ is independent of D_{ i } for stray field skyrmions and scales linearly with D_{ i } for DMI skyrmions. Fourth, DMI skyrmions are rather insensitive to applied fields, while stray field skyrmions are quite sensitive. And fifth, ψ is variable for stray field skyrmions but not for DMI skyrmions.
The collapse radius is a particularly useful and technologicallyrelevant distinguishing feature. As shown in Fig. 2c, DMI skyrmions are those with roomtemperature collapse diameters <10 nm and stray field skyrmions are those collapsing at larger sizes. This observation is supported by similar diagrams as a function of all material parameters, see Supplementary Information. With a very few exceptions (at small thicknesses), the 10 nm threshold is unambiguous even with some experimental error. That is, skyrmions either collapse at much larger or much smaller diameters than 10 nm. Even though further research is required to check if this threshold would change for other classes of materials (e.g., B20 materials or ferrimagnets) and for skyrmion lattices, we conclude that the 10 nm threshold size is a practical and valid criterion to identify the type of a skyrmion experimentally, at least for transition metal ferromagnet multilayers.
Finally, we remark that all room temperature isolated skyrmions observed so far^{11,22,23,24,25,26,27,50,51,52,53,54,55,56} should be classified as stray field skyrmions. Isolated DMI skyrmions have been observed at cryogenic temperatures^{19,57,58}, but not yet at room temperature. The DMI strength measured in sputtered multilayers, however, can be on the order of 2 mJ/m^{2}^{22,23,24,59,60,61,62,63}, which we predict to be sufficient to stabilize DMI skyrmions. But the bistability region can make it difficult to obtain them by just shrinking stripe domains, in particular when pinning is sizable^{56}. From our analysis is seems likely that DMI skyrmions can be observed in existing sputtered multilayer materials by using novel skyrmion generation approaches, such as spinorbit torque nucleation^{51}. To verify that these skyrmions are DMI skyrmions, one only needs to confirm that they can be as small as 10 nm in diameter.
Bistabilities and zero stiffness skyrmions
The phase diagram of skyrmions in materials with strong stray fields shows a richness that has not been explored due to the inherent limitations of existing models. A particularly fascinating example is the appearance of multiple (possibly degenerate) minima in E(R). These minima can be separated by energy barriers exceeding 50k_{ B }T_{RT} at room temperature, indicating that both states can be simultaneously stable. Although the possible coexistence of compact and bubblelike has previously been suggested^{39,49}, concrete model predictions of bistability have only been made in the case of systems stabilized by lateral constraints^{64}. Our model shows that bistability is a more general phenomenon. As depicted in Fig. 3a, bistability exists in a small pocket of the phase diagram near the point where the collapse fields of stray field and DMI skyrmions coincide. The two types of skyrmions in the bistability region can have very different properties (Fig. 3b), confirmed by micromagnetic simulations: Their radii differ by more than one order of magnitude and their spin structure is Néellike for the small (DMI) skyrmion and intermediate for the large (stray field) skyrmion. The different size and domain wall angle can be used to encode information or to move the skyrmions in noncollinear directions by spin orbit torques.
If the energy barrier between the multiple minima is on the order of k_{ B }T or smaller, thermal fluctuations can cause the system to dynamically oscillate between these minima with frequencies on the order of the attempt frequency (reported to be between 2.5 × 10^{7}s^{−1} ^{65} and 4.5 × 10^{−9} s^{−1} ^{47}). We therefore call these states zero stiffness skyrmions. Figure 3c shows E(R) for a system in which the radius can thermally fluctuate between 2 nm and 11 nm, such that it exhibits effectively zero stiffness with respect to variations of radius within this range. Zero stiffness skyrmions exists at larger DMI than bistable skyrmions (see Fig. 2d). We expect that zero stiffness skyrmions have a very low resonance frequency associated with their breathing mode, which could be exploited in nonlinear skyrmion resonators^{66} and should have impact on their inertia^{11} and on skyrmion Hall angle^{52,53}. The ultra short timescale of sizable thermal fluctuations (a factor five in radius) and the resulting randomness of spinorbit torque driven motion could also be used to implement highspeed skyrmion randomizers and stochastic computing^{67}.
Skyrmions for racetrack memory applications
We now consider the design of skyrmions suitable for applications, such as racetracktype memory devices in which bit sequences are encoded by the presence and absence of skyrmions that can be shifted by current^{12,13,14,68}. Three necessary attributes are (i) small bit sizes, (ii) long term thermal stability, and (iii) stability in zero or low applied field. Figure 4 explains why ferromagnetic films and multilayers are incapable of meeting these requirements, and identifies alternative materials that can host sub10 nm zerofield skyrmions at room temperature.
Zero field skyrmions reside at minima of the domain wall energy, with energy barriers \({E}_{a}^{0}\) and \({E}_{a}^{\infty }\) that prevent shrinking to zero size and infinite growth, respectively. We take the smaller of these as the effective energy barrier \({E}_{a}^{{\rm{eff}}}\) to estimate stability. Figures 4a,b show size and stability, respectively, for zerofield skyrmions as a function of K_{ u } and M_{ s } for thin films with D_{ i } = 2 mJ/m^{2}, typical of heavymetal/ferromagnet interfaces. Stray fields tend to reduce \({E}_{a}^{\infty }\) and hence zero field skyrmions occur preferably at low M_{ s }, but there is an inherent tradeoff in Figs 4a,b between stability and size. Moderate applied fields (Figs 4c,d) extend the stability range for large stray field skyrmions but do little to stabilize small skyrmions.
Since it is the domain wall energy whose minimum is responsible for zerofield skyrmions, stability can be enhanced by increasing film thickness since this energy scales with d. For higher M_{ s } materials, the benefit is offset by destabilizing stray fields that lead to stripe domain formation. But at low M_{ s }, sub10 nm skyrmions can readily be achieved, as shown in Figs 4d,e for the limit M_{ s } = 0. These figures show zerofield skyrmion size versus A and D_{ i }, with K_{ u } adjusted at each point to ensure \({E}_{a}^{{\rm{eff}}}=50{k}_{B}{T}_{{\rm{RT}}}\). As a rule of thumb, κ ≈ 0.8 can be used to estimate K_{ u } from the diagram.
Figure 4b shows parameter ranges for several classes of materials, providing guidance to realize such skyrmions experimentally. The problem with thinfilm ferromagnets is that the quality factor \(Q=\frac{2{K}_{u}}{{\mu }_{0}{M}_{s}^{2}}\) is usually constant when engineering multilayers or alloys, which correlates K_{ u } and M_{ s } in a way that makes the lowM_{ s }, highK_{ u } region inaccessible. This is reflected in the diagonal lower boundary marking the accessible parameter range for such materials (red box, Fig. 4b), and in the trend for experimental film parameters (red circles) reported in the literature. This explains why roomtemperature skyrmions observed in ferromagnetic films and multilayers have all been quite large. Note that all previously observed isolated zero field skyrmions in Cobased ferromagnets are only stabilized by pinning or sample boundaries^{23}.
Ferrimagnets, by contrast, generally show little correlation between K_{ u } and M_{ s }, as seen in the representative literature values marked by triangles in Fig. 4b. In rareearthtransition metal alloys, these parameters can be tuned independently by temperature or composition to achieve the combination of lowM_{ s } and high K_{ u } needed for stabilizing small, zerofield skyrmions. Moreover, the low M_{ s } and bulk perpendicular anisotropy in such materials allows for thicker films with strong PMA, which reduces the D_{ i } needed to stabilize small skyrmions (compare Figs 4e,f). Finally, as seen in Figs 4e,f, values for A between 4 pJ/m and 10 pJ/m are ideal, and in the rareearthtransitionmetal ferrimagnets the exchange constants are found to be just in this range. In addition to compensated ferrimagnets, natural antiferromagnets^{69,70,71,72,73} are quite promising for ultrasmall skyrmions, as are Cobased synthetic antiferromagnets (SAFs) that minimize stray fields, where heavymetal interfaces can provide substantial K_{ u } and D_{ i }.
Spinorbit torque driven motion of skyrmions
Finally, we derive analytical expressions for skyrmion dynamics that agree well with micromagnetic simulations and guide material selection to realize currentdriven velocities exceeding 1000 m/s for sub10 nm skyrmions. Figure 5a shows an intermediate skyrmion driven by dampinglike spinorbit torque (SOT) from a charge current density j_{HM} in an adjacent heavy metal. Due to its topological charge, the skyrmion moves at an angle ξ′ with respect to j_{HM}, a phenomenon known as the skyrmion Hall effect^{52,53}. For Néel skyrmions ξ′ coincides with the skyrmion Hall angle ξ, but in general ξ ≠ ξ′ since the currentinduced effective force F depends on domain wall angle ψ.
Figures 5b,c show micromagnetic simulations of SOTdriven motion for a 200 nm and 20 nm skyrmion, respectively, in a ferromagnetic Cobased multilayer. The skyrmion Hall angle and currentdriven mobility μ = v/j_{HM} are plotted versus skyrmion size in Fig. 5d. One sees that for small skyrmions ξ approaches π/2 and the mobility drops dramatically (linearly with radius) so that ultrasmall ferromagnetic skyrmions are effectively immobile. These phenomena have been noted in previous micromagnetic simulations^{50}, but so far neither an intuitive explanation nor a materialsbased solution for this technologicallycritical challenge have been presented.
In the Supplementary Information we use the Thiele equation^{74} to derive the steady state velocity v = v(cosξ′, sinξ′) of a skyrmion driven by spintransfer^{75} and spinorbit torques based on the 360° DW model. Our treatment includes multilayers such as synthetic ferrimagnets or antiferromagnets with multiple sublattices i of variable d_{ i } and M_{s,i}, and can be extended to natural ferrimagnets by suitably scaling γ and α^{83}. The solid lines in Fig. 5d,e show the results of our analytical model applied to ferromagnets and synthetic antiferromagnets, respectively. Although our model cannot account for internal deformations, we find excellent agreement with micromagnetic simulations for skyrmion diameters <200 nm, which distort minimally in micromagnetic simulations and are the most technologically relevant.
For dampinglike SOT the angle between v and j_{HM} is
with
Here α is the Gilbert damping, \({\theta }_{{\rm{SH}}}^{{\rm{eff}}}\) is the effective spin Hall angle, and I_{ A } is the exchange integral (see Supplementary Information). 〈x〉 denotes the average of x over all the layers or sublattices, weighted by dM_{ s } of each layer: 〈x〉 = (∑_{ i }d_{ i }M_{s,i}x)/(∑_{ i }d_{ i }M_{s,i}). The mobility is given by:
Here, the final term represents the summation of the currentinduced effective fields in each layer, with γ the gyromagnetic ratio, ħ the reduced Planck constant and e the electron charge. The first term is a damping term that includes a topological contribution proportional to 〈N〉. For compact skyrmions, \(\tilde{g}(\rho )\approx 0.9\gg \alpha \), i.e., the topological damping term dominates the dynamics. The ratio ΔI_{ D }(ρ)/I_{ A }(ρ) derives from the Thiele effective force, with
For large ρ it is proportional to Δ, whereas for small ρ it is proportional to R, which explains why it is difficult to drive small skyrmions fast with SOT.
Based on these relations, there are only two ways to enhance the mobility of small, compact skyrmions: reduce the (topological) damping and enhance the currentinduced effective field. The former can be accomplished using multisublattice materials and structures, such as ferrimagnets and SAF multilayers. The latter can be achieved using lowM_{ s } materials interfaced with strong spin Hall metals, since the currentinduced effective field scales as \({\theta }_{{\rm{SH}}}^{{\rm{eff}}}/{M}_{s}\), as verified experimentally for Ta/TbCo^{76}.
Figure 5e shows ξ and μ for the bilayer SAF structure in Fig. 5f as a function of the compensation ratio M_{s1}/M_{s2} of the antiferromagneticallycoupled layers, with parameters such that the layers host skyrmions with diameter of 10 nm. One finds an enhancement of the mobility as the system approaches magnetic compensation due to a reduction in the topological damping. Note that the mobility takes a maximum at \({M}_{s2}/{M}_{s1} < 1\) because M_{s1} is fixed in the simulations and M_{s2} is varied. Thus as M_{s2}/M_{s1} decreases from unity, the average M_{ s } also decreases, which initially leads to an increase of μ.
Prior micromagnetic simulations have suggested SAF structures to mitigate the skyrmion Hall effect by reducing the net topological charge. The corresponding reduction in topological damping is the main reason why larger skyrmion velocities are also observed^{70,77}. We note that SOT driven domain walls in SAF structures have been shown to achieve enhanced mobility due to interlayer exchange torque^{78}, a mechanism that is fundamentally different from what we identify here, and which is not effective for skyrmions due to the circular symmetry. Finally, Fig. 5e shows that by decreasing M_{ s } of the constituent layers to also increase the spin Hall effective field per unit current density, velocities exceeding 1000 m/s can be achieved at a reasonable current density of 10^{12}A/m^{2} for 10 nm skyrmions, as required for applications.
Based on the material requirements identified above, (i) a low d_{ i }M_{s,i} in each layer, (ii) a large effective spin Hall angle in each layer, (iii) two (or more) antiferromagnetically coupled layers or sublattices with a low (ideally zero) average topological charge 〈N〉, we can now identify particular materials. We predict that one of the most promising materials is Pt/GdCo/GdFe/Ir/Ta. GdCo/GdFe bilayers can be coupled antiferromagnetically by just tuning one of the layers to be above and one to be below the compensation temperature^{79}. The antiferromagnetic coupling is much stronger than for RKKY coupled multilayers and not sensitive to the layer thicknesses. The material has low M_{ s } and bulk perpendicular magnetic anisotropy, i.e., it can be easily grown to be 5 nm thick in a single layer. Pt and Ir lead to large additive DMI in combination with Co and Fe, respectively^{22,59}. Finally, Pt and Ta are known to be efficient spinorbit torque materials with opposite (additive) sign of the spin Hall angle.
Discussion
In this article, we have presented the first unified theory that describes both stray field and DMI skyrmions in one coherent model. In order to make this model readily available, we have included software implementations in an online repository^{84}.Thanks to the accurate expression of stray field interactions, we have been able to answer a number of fundamental physical questions, such as the meaning of topological stability and the proper and physically justified distinction of DMI and stray field skyrmions. Importantly, only DMI skyrmions can be <10 nm in diameter at room temperature. We have demonstrated that stray field interactions in ferromagnetic materials prevent the formation of such DMI skyrmions at at moderate applied fields and presently accessible values of D_{ i } ≤ 2 mJ/m^{2} and that lowM_{ s } materials are hence required to observe sub10 nm skyrmions at room temperature. Furthermore, we find that small skyrmions in ferromagnets are much slower than in ferrimagnets or (synthetic) antiferromagnets. While this result is similar to regular domain walls, the underlying physics is different: while antiferromagnetic domain walls benefit from an exchange torque^{78}, skyrmions instead are affected by a topological damping that can only be reduced in systems with antiferromagnetically coupled sublattices. Considering the topological damping is hence of crucial imporance for skyrmionrelated materials science. It hence became clear that the future of skyrmionic lies in ferrimagnets and antiferromagnets and that transition metal ferromagnetic multilayers cannot be engineered to meet the requirements of data storage applications. Finally, we note that our model does not treat twisted states that could arise in lowDMI systems with strong stray field interactions^{42}, nor does it treat skyrmions stabilized by more exotic phenomena such as frustrated exchange interactions^{80,81,82}. However, the most suitable materials for applications are strong DMI systems with small, highlymobile skyrmions, and our model accurately predicts all relevant physics for fundamental and applied studies of such skyrmions.
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Acknowledgements
This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award #DESC0012371. FB thanks Alexander Stottmeister, Benjamin Krüger, and Kai Litzius for fruitful discussions and the German Science Foundation for financial support under grant number BU 3297/11.
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F.B. conceived the project, derived the energy functions, generated and analyzed the data, and drafted the manuscript. I.L. derived the analytic expression for ψ. GSDB supervised the project. All authors discussed the results, the implications, and the figures and commented on the manuscript.
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Büttner, F., Lemesh, I. & Beach, G.S.D. Theory of isolated magnetic skyrmions: From fundamentals to room temperature applications. Sci Rep 8, 4464 (2018). https://doi.org/10.1038/s41598018222428
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