Abstract
Transitionmetal interfaces and multilayers are a promising class of systems to realize nanometersized, stable magnetic skyrmions for future spintronic devices. For room temperature applications, it is crucial to understand the interactions which control the stability of isolated skyrmions. Typically, skyrmion properties are explained by the interplay of pairwise exchange interactions, the DzyaloshinskiiMoriya interaction and the magnetocrystalline anisotropy energy. Here, we demonstrate that higherorder exchange interactions – which have so far been neglected – can play a key role for the stability of skyrmions. We use an atomistic spin model parametrized from firstprinciples and compare three different ultrathin film systems. We consider all fourthorder exchange interactions and show that, in particular, the foursite four spin interaction has a large effect on the energy barrier preventing skyrmion and antiskyrmion collapse into the ferromagnetic state. Our work opens perspectives to stabilize topological spin structures even in the absence of DzyaloshinskiiMoriya interaction.
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Introduction
Magnetic skyrmions—localized spin structures with a topological charge^{1}—have raised high hopes for future magnetic memory and logic devices due to their nanoscale dimensions, stability, and ultralow energydriven motion^{2,3,4,5,6}. Skyrmion lattices have been first observed in bulk magnets with a broken inversion symmetry in their crystal structure^{7,8}. The discovery of a skyrmion lattice in a single atomic layer of Fe on the Ir(111) surface^{9} has opened the door to a new class of systems: transitionmetal interfaces and multilayers. Due to the possibility of varying film composition and structure, these systems allow to modify magnetic interactions and thereby the properties of skyrmions. At such transitionmetal interfaces, individual magnetic skyrmions with diameters ranging from a few 100 nanometers down to a few nanometers have been realized as a metastable state in the fieldpolarized ferromagnetic background^{8,10,11,12,13,14,15,16} as needed for applications.
A key challenge of skyrmion based data processing and storage technology is the robustness of information carriers, i.e., stability of the skyrmionic bits, against random thermal fluctuations at operating temperatures. At finite temperature, the magnetic moments of skyrmions are coupled to the environment, which induces fluctuations. Over time, a rare energy fluctuation can grow in excess of the barrier height and can prompt the skyrmion to overcome the barrier and collapse to the ferromagnetic background leading to a loss of topological charge. Therefore, an accurate assessment of barrier height is essential to determine the stability of skyrmions. To achieve data reading and writing capabilities of skyrmionic bits with high efficiency, a control over the barrier height is also necessary.
The existence of chiral magnetic skyrmions^{17} is ascribed to a competition of the Heisenberg pairwise exchange interaction, the DzyaloshinskiiMoriya interaction (DMI)^{18,19}, the magnetocrystalline anisotropy, and the dipoledipole interactions. A prerequisite of DMI—which provides a unique rotational sense to skyrmions—is the concerted action of spinorbit coupling and broken inversion symmetry, which can be achieved at interfaces of transitionmetals^{20}. The DMI further stabilizes metastable isolated skyrmions against annihilation into the ferromagnetic background^{17,21}. Often the exchange interactions are treated in a micromagnetic or effective nearestneighbor approximation. However, the exchange interactions are longrange in itinerant magnets such as 3d transitionmetals. This can lead to a competition between exchange interactions from different shells of atoms resulting in an enhanced skyrmion stability^{16,22} even in the absence of DMI^{23}.
The itinerant character of 3d transitionmetals limits the applicability of the Heisenberg model to describe their magnetic properties. Based on the spin1/2 Hubbard model, it has been shown that the higherorder exchange interactions (HOI) beyond pairwise Heisenberg exchange can arise such as the twosite four spin (biquadratic) or the foursite four spin interaction^{24,25}. Such higherorder terms can lead to intriguing magnetic ground states due to a superposition of spin spirals—socalled multiQ states—which have been predicted based on firstprinciples calculations^{26}. The interplay of the foursite four spin interaction and DMI is the origin of the nanoskyrmion lattice of the Fe monolayer on Ir(111)^{9} and the effect of the biquadratic interaction on skyrmion lattice formation has been studied systematically^{27}. It has been further demonstrated that the HOI can compete with the DMI and stabilize novel magnetic ground states^{28}. Based on a multiband Hubbard model, a threesite four spin interaction has recently been proposed for systems with a spin beyond S = 1/2 in addition to the biquadratic and the foursite four spin interaction in fourthorder perturbation theory of the hopping parameter t with respect to the Coulomb energy U^{29}. This term has been attributed to stabilize a doubleQ or socalled upupdowndown (uudd) state in an Fe monolayer on Rh(111)^{30}. Despite the compelling experimental evidence of the relevance of HOI^{9,28,30,31,32}, they have been neglected so far in the theoretical description of the properties of isolated magnetic skyrmions at transitionmetal interfaces.
Here, we reveal the intriguing role played by the HOI for the stability of topologically nontrivial spin structures such as skyrmions and antiskyrmions at transitionmetal interfaces. We use spin dynamics simulations based on an atomistic spin model with all parameters calculated via density functional theory (DFT). The energy barrier, preventing skyrmions and antiskyrmions to collapse into the ferromagnetic state, is obtained using the geodesic nudged elastic band (GNEB) method^{33}. We consider three ultrathin film systems: (i) fccPd/Fe bilayer on Rh(111) for which sub10 nm skyrmions have been predicted at low magnetic field^{34}, (ii) fccPd/Fe bilayer on Ir(111), the most intensively studied ultrathin film system that hosts isolated skyrmions^{10,11,22,35,36,37,38,39,40,41,42} and an hcpFe/Rh bilayer on Re(0001) with an inplane easy magnetization axis^{43}.
Upon including the HOI, the stability of skyrmions and antiskyrmions in all of these films is greatly modified. Surprisingly, the effect of the biquadratic and the threesite four spin interaction concerning the energy barrier is to a good approximation already captured in the exchange constants obtained by mapping the DFT results to a spin model neglecting HOI. The foursite four spin interaction has a large effect on the saddle point and is responsible for the large change in energy barriers. We find a linear scaling of the barrier height with the foursite four spin interaction. The barrier is enhanced or reduced depending on its sign. Even small values of the foursite four spin interaction of 1–2 meV, typical for 3d transition metals, modify the energy barrier by 50–120 meV. This leads to a huge enhancement or reduction of the skyrmion or antiskyrmion lifetime. We further show that the HOI can stabilize topological spin structures in the absence of DMI.
Results
Atomistic spin model and DFT calculations
We describe the magnetic state of an ultrathin film by a set of classical magnetic moments {M_{i}} localized on each atom site i of a hexagonal lattice and their dynamics is governed by the following Hamiltonian:
where m_{i} = M_{i}/M_{i} is a unit vector. The exchange constants (J_{ij}), the DMI vectors (D_{ij}), the magnetic moments (μ_{s}) and the uniaxial magnetocrystalline anisotropy energy (MAE) constant (K) were calculated based on DFT (see refs. ^{22,34,39,43}). We neglect the dipoledipole interaction since it is small in ultrathin films, which is of the order of 0.1 meV per atom, and it can be effectively included into the MAE^{44,45}.
The last three terms are the biquadratic interaction (B_{ij}), the threesite four spin interaction (Y_{ijk}) and the foursite four spin interaction (K_{ijkl}), respectively. Since these terms arise from the fourthorder perturbation theory, we restrict ourselves to the nearestneighbor approximation, i.e., up to the first term of these HOI. The corresponding constants are denoted as B_{1}, Y_{1} and K_{1}. The evaluation scheme of the HOI on a hexagonal twodimensional (2D) lattice is illustrated in Fig. 1b.
First, we calculate the energy dispersion of homogeneous flat spin spirals via DFT without taking spinorbit coupling (SOC) into account to determine the exchange constants. A spin spiral is characterized by a wave vector q in the 2D Brillouin zone (2DBZ, Fig. 1d) and the magnetic moments of an atom at lattice site R_{i} is given by \({{\bf{M}}}_{i}=M(\sin ({{\bf{qR}}}_{i}),\cos ({{\bf{qR}}}_{i}),0)\) with the size of the magnetic moment M. Fig. 1e shows the energy dispersion E(q) of spin spirals without SOC along two high symmetry directions obtained via DFT for an fccPd/Fe bilayer on the Rh(111) surface^{34} (Fig. 1a). At the high symmetry points of the 2DBZ, we find wellknown magnetic states: the ferromagnetic (FM) state at the \(\overline{\Gamma }\) point, the rowwise antiferromagnetic (AFM) state at the \(\overline{{\rm{M}}}\) point and the Néel state with angles of 120° between adjacent magnetic moments at the \(\overline{{\rm{K}}}\) point.
Clearly, the FM state is energetically lowest among these three states (Fig. 1e). Along both high symmetry directions, the 90° spin spirals (Fig. 1c) are found at \({\bf{q}}=(1/2)\overline{\Gamma {\rm{M}}}\) and at \({\bf{q}}=(3/4)\overline{\Gamma {\rm{K}}}\) (Fig. 1d). The total energy of homogeneous spin spirals without SOC is fitted to functions obtained by expressing the Heisenberg model (first term of Eq. (1)) in reciprocal space to extract the exchange interaction parameters up to 11th nearest neighbors (Supplementary Table 1).
Since the functional form of the threesite four spin and the biquadratic interactions for homogeneous spin spirals resemble that of the first three exchange constants, we cannot separate the exchange and the higherorder constants by fits (see “Methods”). Therefore, we calculate the HOI constants from the energy difference between the spin spiral (singleQ) and multiQ states without SOC and modify the exchange constants obtained from a fit to only the first term of Eq. (1).
The three selected multiQ states are a superposition of spin spirals (neglecting SOC) corresponding to symmetry equivalent q vectors (Fig. 1c, d) and have a constant magnetic moment at every atomic site. Within the Heisenberg model of pairwise interaction, the multiQ and singleQ states are energetically degenerate. However, the HOI lift the degeneracy which provides a way to compute their strengths. In DFT calculations, all the interactions are implicitly included through the exchangecorrelation functional. Therefore, we can obtain the HOI constants from total energy calculations of multiQ and singleQ states without SOC.
We consider two collinear states, the socalled uudd or doublerow wise antiferromagnetic states^{46} and a threedimensional noncollinear state, the socalled 3Q state^{26}, to uniquely determine three higherorder exchange constants (for spin structures see insets of Fig. 1e). The biquadratic (B_{1}), the threesite four spin (Y_{1}) and the foursite four spin interaction (K_{1}) constants are computed from the energy differences between the multiQ and the corresponding singleQ states (without SOC) by solving the equations^{29}:
The three multiQ states are higher in energy compared to the corresponding spin spiral states without SOC for Pd/Fe/Rh(111) (Fig. 1e) and far above the FM state (for energies see Supplementary Table 2). Nevertheless, they have a large effect on skyrmions in this film system as we show below.
The computed HOI constants modify the first three exchange constants, obtained from fits of the spin spiral energy dispersion neglecting HOI (see “Methods” for a derivation):
where we denote the exchange parameters obtained from fits neglecting HOI as unprimed and the modified ones by considering the higherorder terms as primed. Note the special role played by the foursite four spin interaction which does not adjust any exchange parameter since its contribution to the energy dispersion of spin spirals is a constant value of −12K_{1} independent of the spin spiral vector. Further the HOI only modify the first three exchange constants and, therefore, the other exchange constants used in atomistic spin dynamics simulations remain unchanged (Supplementary Table 1).
The first three exchange and the higherorder exchange constants of Pd/Fe/Rh(111) are displayed in Table 1 along with those for Pd/Fe/Ir(111) and Fe/Rh/Re(0001) obtained by similar DFT calculations (see also Supplementary Table 2). We find that the Pd/Fe bilayer on Rh(111) and on Ir(111) behaves similarly in terms of exchange and higherorder exchange constants since Rh and Ir are isoelectronic 4d and 5d transition metals. In these two film systems, the signs of the nearestneighbor exchange constant (J_{1}) and the second and third nearest neighbors are opposite which leads to exchange frustration^{22,34}. The exchange interaction in Fe/Rh/Re(0001), in contrast, is dominated by the nearestneighbor exchange constant. Note that the sign of the biquadratic (B_{1}) and the foursite four spin constants (K_{1}) is negative in Fe/Rh/Re(0001), while it is positive for the other two systems. As shown below, the sign of K_{1} is essential for the skyrmion stability in these films.
SOC introduces two additional contributions: DMI and MAE (see “Methods” for details). The DMI lowers the energy of cylcoidal spin spirals with a clockwise rotational sense in the vicinity of the \(\overline{\Gamma }\) point (see inset of Fig. 1e). The MAE shifts the energy of spin spirals by K/2 with respect to the FM state. The DMI constants and MAE are given in Supplementary Table 3. Note that, very recently, higherorder interactions arising from SOC have been proposed for transition metal systems^{32,47,48}. However, experimental evidence for these interactions is missing and we exclude them from our current investigation.
Spin dynamics simulations
We use atomistic spin dynamics simulations (see “Methods”) based on the Hamiltonian of Eq. (1) with all parameters obtained from DFT including DMI, MAE and total magnetic moments, as discussed in the previous section, to calculate the zero temperature phase diagram and the properties of isolated skyrmions and antiskyrmions in the three film systems. The results for Pd/Fe/Rh(111) are shown in Fig. 2 (for the other systems see Supplementary Figs. 1 and 2).
We first discuss the effect of the HOI on the zero temperature phase diagram (Figs. 2a, b). The FM state and the homogeneous spin spiral states remain almost unaffected by the higherorder terms. However, the skyrmion lattice loses a small amount of energy with respect to the homogeneous spin spirals which remains constant throughout the range of magnetic fields. This leads to an expansion of the spin spiral and FM phases at the expense of the skyrmion lattice phase, which is squeezed. Since isolated skyrmions can be stabilized in the FM (fieldpolarized) phase, it is of prime importance. The onset of the FM phase, characterized by the critical field B_{c}, has shifted from 2.75 T to a lower value of 2.25 T due to the HOI. As expected from the magnetic interaction constants, a similar trend of the phase diagram is obtained for Pd/Fe/Ir(111) (Supplementary Fig. 1). However, since the higherorder constants are quite small for Fe/Rh/Re(0001) (cf. Table 1), the phase diagram is basically unchanged (Supplementary Fig. 2).
In our spin dynamics simulations, we have created isolated skyrmions and antiskyrmions in the fieldpolarized background following the theoretical profile^{49} and relaxed the spin structures with the full set of DFT parameters. The radius of skyrmions and antiskyrmions—defined as in ref. ^{49}— increases with HOI for Pd/Fe/Rh(111) on average by ~35% and 30%, respectively (Fig. 2g). The skyrmion and antiskyrmion profiles at 5.2 T (see insets of Fig. 2g) can be fitted by the standard skyrmion profile. The antiskyrmion exhibits a steeper profile which reflects that it has a smaller radius than the skyrmion. Similar trends of the skyrmion and antiskyrmion radii are found for Pd/Fe/Ir(111) (Supplementary Fig. 1). Due to relatively small HOI, the skyrmion radii remain almost unchanged for low magnetic fields above B_{c} for Fe/Rh/Re(0001) (Supplementary Fig. 2 and Supplementary Note 1).
To study the stability of metastable isolated skyrmions and antiskyrmions, we calculate the minimum energy path (MEP) for the collapse of a single skyrmion or antiskyrmion into the FM background (see “Methods”). The point of maximum energy on this path, known as the saddle point, with respect to the initial state (skyrmion or antiskyrmion) is a measure of the barrier height. As seen in Fig. 2h, the HOI increase the energy barrier for skyrmion annihilation in Pd/Fe/Rh(111) by more than a factor of two at small magnetic fields above B_{c}. For antiskyrmions, the barrier height is even increased by a factor of 5.
The energy barriers of skyrmions vary nonlinearly at small magnetic fields, and, thereafter, reduce almost linearly with increasing magnetic fields^{22}. On an average, we notice an increase in barrier height of nearly 100 meV for skyrmions upon including the HOI. At low fields up to B − B_{c} = 1.36 T, there is a transition from the normal radial collapse mechanism^{22,34} without HOI to a chimera collapse mechanism^{16,50} with HOI. Above B − B_{c} = 1.36 T, the skyrmions merge into the FM background through the normal radial collapse without HOI, which remains unchanged after including HOI.
The barrier heights of antiskyrmions with HOI exhibit a similar variation with field as that of skyrmions. The energy barriers of antiskyrmions without HOI are extremely small (~20 meV) which implies that they are basically unstable even at cryogenic temperatures. However, after including HOI, the energy barriers become ~100 meV at small fields (up to 1.5 T above B_{c}), which suggests that metastable antiskyrmions could be realized in experiments. The annihilation mechanism of antiskyrmions is via the radial collapse^{22}, which is unaffected upon including HOI.
In Pd/Fe/Ir(111), the HOI increase the energy barriers of isolated skyrmions and antiskyrmions by similar values of ~100 and ~90 meV, respectively (Supplementary Fig. 1). On the other hand, in Fe/Rh/Re(0001), the stability of isolated skyrmions is reduced on an average by 70 meV upon including the HOI (Supplementary Fig. 2). This large barrier reduction shows that even small values of the higherorder constants (cf. Table 1) can have significant effects.
Analysis of collapse mechanisms
Now we focus on the question which of the HOI is responsible for the large changes of the energy barriers for skyrmion or antiskyrmion collapse. We consider both collapse mechanisms for skyrmions, i.e., the chimera collapse at low fields and the radial collapse at higher fields and the radial collapse mechanism for antiskyrmions (cf. Fig. 2h). In Fig. 3, the energy decomposition of three representative minimum energy paths are displayed at selected magnetic fields for Pd/Fe/Rh(111) with and without taking HOI into account (for the other two systems see Supplementary Figs. 3 and 4).
The total energy rises along the MEP as one moves from the initial (skyrmion) state to the saddle point and descend thereafter to the final (ferromagnetic) state (Fig. 3a). In the simulation neglecting the HOI, we find that the energy barrier is dominated by the energy contribution from the DMI, which favors the skyrmion state. Due to exchange frustration, there is also a small energy contribution to the barrier from the exchange energy. Naturally, the energy due to the Zeeman term and the magnetocrystalline anisotropy decrease in the ferromagnetic state.
Upon including the HOI (Fig. 3b), we find the large increase of the energy barrier as discussed in the previous section. In addition, the annihilation mechanism changes at this magnetic field from the radial collapse without HOI (Fig. 3a) to the chimera collapse mechanism with HOI (cf. Fig. 3j, k which show the spin structures in the vicinity of the saddle point of the two types of annihilation mechanisms). Interestingly, the chimera collapse mechanism has been previously discussed in ultrathin films with very strong exchange frustration^{16,50,51}, which suggests that HOI act in a similar way. The energy decomposition shows that the DMI contribution is of similar magnitude at the saddle point of the path with and without HOI (Fig. 3c). However, one cannot compare the exchange interactions before and after the HOI are included in the simulations, since the higherorder terms modify the exchange constants according to Eqs. (5)–(7). Therefore, we add the contributions due to the exchange, the threesite four spin interaction and the biquadratic terms which we denote as combined exchange. The comparison of Fig. 3a, b shows that the exchange and the combined exchange behave qualitatively quite similar along the path—which is also true for the other two collapse mechanisms (Fig. 3d, e and Fig. 3g, h). The absolute energy change from exchange to combined exchange at the saddle point is relatively small (Fig. 3c, f, i).
The foursite four spin interaction acts in a qualitatively different way compared to all other terms. For all considered paths (Fig. 3b, e, h), it gains in energy slowly as one approaches the saddle point, in the vicinity of the saddle point it becomes very steep, reaches a maximum at the saddle point and drops quickly thereafter. The energy contributions at the saddle point (Fig. 3c, f, i) show that it provides by far the largest difference between the simulations with and without HOI, irrespective of the collapse mechanism or the initial state.
The energy contribution from the threesite four spin and the biquadratic interactions decreases along all collapse processes and the energy drop escalates after the saddle point (see insets of Fig. 3b, e, h). The difference in energy profile of the DMI and MAE is an associated effect of the HOI caused by the changes in relative spin angles during the collapse process. Fig. 3c, f, i show that the combined exchange can provide a tiny contribution to the energy barrier depending on the collapse mechanism, while the DMI and Zeeman terms assert only a little weight if not compensated by each other. Therefore, the foursite four spin interaction mainly controls the change of the barrier height.
The spin structure in the vicinity of the saddle point is shown in Figs. 3j, k for the chimera and the radial collapse of the isolated skyrmion including the effect of HOI. We see that the radial collapse of an isolated skyrmion is very similar to that found by neglecting HOI in ref. ^{22}. However, at the saddle point, there are four spins pointing towards each other while previously a threespin structure was reported. The unusual saddle point including HOI is obtained throughout the studied field range and for annihilation of skyrmions in Pd/Fe/Ir(111). However, for the skyrmion collapse in Fe/Rh/Re(0001), a threespin structure at the saddle point similar to that of ref. ^{22} occurs. The chimera skyrmion collapse and the radial antiskyrmion collapse are similar to that found in simulations neglecting HOI^{16,22,50}.
We have also performed atomistic spin simulations for Pd/Fe bilayers on Rh(111) and on Ir(111) without and with higherorder interactions of the MEP for the escape mechanism (Supplementary Fig. 5) introduced previously by Bessarab et al.^{38}. The saddle point along this MEP resembles a slightly deformed skyrmion (Supplementary Fig. 5) and is distinctively different from the saddle point of the collapse mechanisms with large spin rotations on the atomic scale. Since the latter property is decisive for the large energy contribution of the foursite four spin interaction, there is almost no influence of HOI on the energy barrier of the escape mechanism.
Analysis of the foursite four spin interaction
To understand the prominent effect of the foursite four spin interaction on the energy barrier, we present its siteresolved energy at the saddle point with respect to the initial state (skyrmions or antiskyrmions) for the three MEPs of Fig. 3b, e, h in Fig. 4a–c. We notice that a group of only 14 spins around the core provide contributions to the foursite four spin interaction, while the surrounding spins do not add any significant value. This finding is independent of whether we consider the saddle point of the chimera collapse (Fig. 4a), the radial skyrmion collapse (Fig. 4b) or the radial collapse of the antiskyrmion (Fig. 4c). Similar observations are made for the other film systems (Supplementary Figs. 6–9). Therefore, the foursite four spin interaction at the saddle point exhibits a general behavior irrespective of the type of collapse mechanism or the initial spin configuration.
In order to explain this localized energy gain at the saddle point, we use a simplified model in which we consider only the site with the largest contribution at the origin. To evaluate the foursite four spin interaction, we need to consider at least the 6 nearest neighbors and the 6 nextnearest neighbors of the central site [cf. Fig. 4d–f]. To simplify the discussion, we slightly symmetrize the spin structure. For the radial skyrmion and antiskyrmion collapse, the 12 neighboring spins are nearly all inplane (Fig. 4e, f). We neglect any outofplane component as shown in Fig. 4h, i to calculate the contributions from the 12 diamonds for the foursite four spin interaction.
For the saddle point of the radial skyrmion collapse (Fig. 4h), we find three distinct types of diamonds which contribute to the foursite four spin interaction. We find a pair of diamonds with values +K_{1} and −K_{1}, two pairs of diamonds with values \(+\frac{1}{2}{K}_{1}\) and \(\frac{1}{2}{K}_{1}\), which cancel out mutually. Out of the six remaining diamonds, there are three groups each containing two diamonds with values \(\frac{\sqrt{3}}{2}{K}_{1}\), −K_{1} and \(+\frac{1}{2}{K}_{1}\), which results in a total energy at the saddle point of \({E}_{{\rm{SP}}}^{{\rm{ISk}}}=2.73{K}_{1}\).
For the saddle point of the radial antiskyrmion collapse (Fig. 4i), we identify three types of diamonds with the same magnitude as for the skyrmion saddle point (Fig. 4h). However, two uncompensated diamonds with \(+\frac{1}{2}{K}_{1}\) and \(\frac{\sqrt{3}}{2}{K}_{1}\) and two diamonds with − K_{1} each lead to a total energy of \({E}_{{\rm{SP}}}^{{\rm{IASk}}}=2.37{K}_{1}\).
The spin structure at the saddle point of the chimera collapse (Fig. 4d) is more complex. There are nonnegligible outofplane components of the spins surrounding the central spin which we take into account in the symmetrization (Fig. 4g). As a consequence, we find six distinct types of diamonds (Fig. 4g). Similar to the other two saddle points, there is a mutual cancellation of many terms which leads to a total energy contribution of the foursite four spin interaction of \({E}_{{\rm{SP}}}^{{\rm{chimera}}}=2.37{K}_{1}\).
Note that the values of these three energies taking the exact spin structure at the saddle points are \({E}_{{\rm{SP}}}^{{\rm{ISk}}}=2.0{K}_{1}\), \({E}_{{\rm{SP}}}^{{\rm{IASk}}}=2.2{K}_{1}\) and \({E}_{{\rm{SP}}}^{{\rm{chimera}}}=2.14{K}_{1}\), which are very close to those obtained using the simplified spin structures.
We obtain similar values for Pd/Fe/Ir(111) at the saddle points corresponding to the skyrmion and antiskyrmion initial states and at the chimera saddle point, the value is only slightly different \({E}_{{\rm{SP}}}^{{\rm{chimera}}}=2.58{K}_{1}\) (Supplementary Figs. 6–8). For Fe/Rh/Re(0001), we find \({E}_{{\rm{SP}}}^{{\rm{ISk}}}=\sqrt{3}{K}_{1}\) (Supplementary Fig. 9).
The energy contribution per site of the foursite four spin interaction for the ferromagnetic state or any flat spin spiral is −12K_{1}, which is also relatively close to the energy in the skyrmion state (cf. Fig. 3b and Supplementary Figs. 3b, e and 4b). Therefore, we obtain an energy difference of E_{SP} − E_{FM} ≈ 10K_{1} for the two symmetric central sites of the saddle point. The surrounding sites provide smaller contributions, however, they still scale with K_{1}. In total, we find an energy contribution of the foursite four spin interaction to the energy barrier of roughly 40K_{1}. Due to the linear dependence on K_{1}, it is also clear that the sign of the foursite four spin interaction determines whether there is an energy gain (K_{1} > 0) or loss (K_{1} < 0) at the saddle point as observed for the two types of systems: Pd/Fe on Rh(111) and on Ir(111) vs. Fe/Rh/Re(0001) (cf. Table 1).
Discussion
Our simplified model states that the barrier height E_{SP} − E_{ISk/IASk} varies linearly with the magnitude of the foursite four spin constant K_{1}. To verify this prediction, we have carried out spin dynamics simulations for three ultrathin film systems at a given magnetic field by changing only the foursite four spin interaction constant while leaving all other interactions the same. Note that the foursite four spin interaction does not affect the energy dispersion of spin spirals which is essential for the equilibrium properties of skyrmions and antiskyrmions. Figure 5 shows that—as expected from our model—the barrier heights for skyrmions and antiskyrmions exhibit a linear scaling with the foursite four spin constant. Only for Pd/Fe/Rh(111), we find a slight deviations from the linear dependence. The scaling constant α, defined as the ratio of the change in energy barrier to the change in foursite four spin constant, is the same for skyrmions and antiskyrmions consistent with our discussion of the energy contributions at the saddle point. We find a value of the scaling constant α of ~40–60 depending on the system.
Figure 3 implies that the HOI can stabilize skyrmions and antiskyrmions by themselves. To test this notion, we have performed spin dynamics simulations by completely switching off the DMI, i.e., setting the DMI from the DFT calculation to zero, while keeping all other magnetic interactions as before. As shown in Fig. 6a, we find stable skyrmions and antiskyrmions for Pd/Fe bilayers on Rh(111) with the same radius which is reduced to 1.4 nm just above B_{c} compared to the case with DMI (cf. Fig. 2g). Due to vanishing DMI, clockwise and anticlockwise rotating skyrmions are degenerate. Large energy barriers of up to 90 meV at B_{c} are obtained (Fig. 6b) due to the foursite four spin interaction, which are identical for skyrmions and antiskyrmions as the DMI is zero.
The decomposition of the energy contributions along the MEP (Fig. 6c), which is the same for skyrmions and antiskyrmions, shows that the sum of all exchange interactions, the biquadratic, and the threesite four spin interaction (combined exchange) results only in a minimal energy barrier. In contrast, the foursite four spin interaction provides the energy barrier and exhibits its characteristic peaklike curve along the MEP. Although the initial skyrmion or antiskyrmion state exhibits a reduced diameter, the energy barrier due to the foursite four spin interaction is almost the same as in the case with DMI (cf. Fig. 3e, h). The saddle point configuration of the skyrmion (Fig. 6d) and antiskyrmion (Fig. 6e) show the characteristic spin structure with four spins at angles of nearly 90° with respect to each other which was observed including DMI (cf. Fig. 3e, f).
As shown in Supplementary Fig. 10, we obtain similar results for Pd/Fe/Ir(111) upon setting the DMI to zero. This suggests that it is possible to stabilize both types of topological states with arbitrary rotational sense due to HOI at inversion symmetric transitionmetal interfaces. Note that it was previously proposed that strong frustration of exchange interactions could stabilize skyrmions and antiskyrmions without DMI. However, a specific ratio between different pairwise exchange interactions is required^{23}.
The lifetime τ of skyrmions or antiskyrmions is given by the Arrhenius law τ = τ_{0}exp(ΔE/k_{B}T), where ΔE is the energy barrier and τ_{0} is the prefactor. Typically, the lifetime is dominated by the energy barrier due to the exponential term. Recently, it has been reported that due to entropy the prefactor can vary drastically with external parameters, e.g., the magnetic field^{52,53}. For bulk magnetic materials a change by 30 orders of magnitude^{52} and for ultrathin films a variation of up to seven orders of magnitude have been found^{53}. However, the effect depends on details of the magnetic interactions. For fccPd/Fe bilayers on Ir(111) and Rh(111)^{34} considered here, there is almost no change of the prefactor and the lifetime is governed by the energy barrier.
An enhancement of the barrier height, ΔE, by 100 meV, as observed for an isolated skyrmion in Pd/Fe/Rh(111), leads to an increase of skyrmion lifetime by orders of magnitude because of the exponential factor. For example, at a temperature of T = 10 K, at which the spinpolarized scanning tunneling microscopy experiments on such ultrathin films are typically performed^{10,11}, we find an enhancement by 50 orders of magnitude, at 100 K, it is still a factor of about 10^{5}, and even at room temperature, it is a factor of about 50. One can also discuss the effect of the HOI in terms of the temperaturedependent phase diagram of Pd/Fe/Rh(111)^{34}. Without the HOI, skyrmions can be stable for an hour up to temperatures of 25 K^{34}, which is increased to a temperature of about 50 K upon including HOI. For antiskyrmions, the change from a barrier of below 20 to ~100 meV leads to an enhancement of their lifetime, which should allow their experimental discovery at least at cryogenic temperatures.
We have demonstrated that higherorder interactions beyond pairwise Heisenberg exchange can play a key role for the stability of skyrmions or antiskyrmions at transitionmetal interfaces. While the biquadratic and threesite four spin interaction contribute in a similar fashion as pairwise exchange interactions to the minimum energy path, we find a qualitative difference for the foursite four spin interaction. Due to the cyclic hopping on four sites it acts on the atomic scale. Therefore, it affects the saddle point of the collapse path strongly while its energy contribution to the initial (skyrmion) and final (ferromagnetic) state is rather similar, in particular, for relatively large skyrmions. This leads to a characteristic peaklike shape of its energy contribution along the collapse path which is otherwise only obtained from a concerted interplay of interactions. On the other hand, DMI provides only an energy difference between the skyrmion and ferromagnetic state. In this respect, the foursite four spin interactions plays a unique role among all considered interactions.
Depending on the sign of the foursite four spin interaction, the energy barrier preventing the collapse of a metastable topological spin structure can be greatly enhanced or reduced. Even for the small values of HOI typical for 3d transition metals, we find large changes of the energy barriers and therefore giant effects on the lifetime, which means that these interactions cannot be neglected. The energy barriers are so much enhanced due to HOI that isolated skyrmions and antiskyrmions can be stable in the absence of DzyaloshinskiiMoriya interaction. Our study opens up another avenue to stabilize topological spin structures at transitionmetal interfaces.
Methods
Firstprinciples calculations
The computational details for calculating the exchange, the DMI, the MAE constants and the magnetic moments are shown in ref. ^{34} for Pd/Fe/Rh(111), in ref. ^{22} for Pd/Fe/Ir(111) and in ref. ^{43} for Fe/Rh/Re(0001). Here, we have evaluated the higherorder exchange constants for all three systems. The electronic structure was calculated using a spinpolarized DFT code based on the projected augmented wave (PAW) scheme^{54} as implemented in the Vienna ab initio simulation package (VASP)^{55}. It ranks among the best available DFT codes in terms of accuracy and efficiency^{56}. We use the same structural parameters as mentioned in the above references. We have used two atomic overlayers on top of nine substrate layers to mimic the surfaces. To maintain consistency with spin spiral calculations, we have chosen local density approximation (LDA) for the exchange and correlation part of potential^{57}. A high energy cutoff of 400 eV was used to precisely calculation the energy of the multiQ states. The 2DBZ was sampled by a MonkhorstPack^{58} mesh of 22 × 28 × 1 kpoints for the uudd state in the \(\overline{\Gamma {\rm{K}}}\) direction, of 14 × 44 × 1 kpoints for the uudd state in the \(\overline{\Gamma {\rm{M}}}\) direction and of 15 × 15 × 1 kpoints for the 3Q state at the \(\overline{{\rm{M}}}\) point. The total energy calculations for the multiQ states were performed without considering SOC and the convergence criteria were set to 10^{−6} eV for all calculations.
Fitting function for HOI
The spin spiral is the exact solution of the classical Heisenberg model for a periodic lattice. The spin spiral, which is characterized by a wave vector q in the 2DBZ and the magnetic moments of an atom at lattice site R_{i}, is given by^{59},
where R_{q} and I_{q} are two vectors that span the xyplane. They obey the following relation,
where M is the magnitude of M_{i} and without loss of generality, we set its norm to unity. The spins of a spin spiral rotate around the zaxis in the xyplane as one moves from one lattice site to another in the direction of q. Using the above two equations, the scalar product of a pair of spins can be written as^{59},
In reciprocal space, q is defined as q = q_{x}b_{1} + q_{y}b_{2}, with b being the reciprocal lattice vectors. In our case, \({{\bf{b}}}_{1}=(2\pi /a)(1,1/\sqrt{3})\) and \({{\bf{b}}}_{2}=(2\pi /a)(1,1/\sqrt{3})\), here a is the inplane lattice constant. For a spin spiral propagating along \(\overline{\Gamma {\rm{KM}}}\), q = (2π/a)q(1, 0) with q ∈ [0, 1] and along \(\overline{\Gamma {\rm{M}}}\)\({\bf{q}}=(2\pi /a)q(\sqrt{3}/2,1/2)\) with \(q\in [0,1/\sqrt{3}]\).
Using Eq. (10) and the first term of Eq. (1), we calculate the energies of the exchange interactions up to 11th nearestneighbor (Supplementary Table 1). In the following, we only show the energy dispersion relation for exchange interactions up to third nearestneighbor and the HOI on a hexagonal lattice along \(\overline{\Gamma {\rm{K}}}\) direction as,
and along \(\overline{\Gamma {\rm{M}}}\) direction as,
It is clear from Eq. (11) and (12) that we can at most obtain the combined terms \({J}_{1}^{\prime}+{Y}_{1}\), \({J}_{2}^{\prime}+{Y}_{1}\) and \({J}_{3}^{\prime}+{B}_{1}/2\) by fits. Therefore, we evaluate the higherorder exchange constants from the total energy difference between the multiQ states and singleQ states according to Eqs. (2)–(4) and modify the exchange constants obtained assuming vanishing higherorder contributions using Eqs. (5)–(7) of the main text. The foursite four spin energy is constant, i.e, −12K_{1}, for all the spin spiral vector q, which implies that it does not affect the exchange constants.
The DMI is calculated in our DFT approach in firstorder perturbation theory on selfconsistent spin spiral states. To obtain the DMI constants, we fit the SOC corrections to the energy dispersion to the second term of Eq. (1). The MAE is calculated from DFT energy differences of the inplane and outofplane magnetization directions obtained in the second variation approach^{60}.
Atomistic spin dynamics simulations
We study the time evolution of atomistic spins as described by LandauLifshitz equation, where the dynamics is expressed as a combination of the precession and the damping terms:
where ℏ is the reduced Planck constant, α is the damping parameter and the Hamiltonian is H defined in Eq. (1). In the simulation, α has been varied from 0.05 to 0.1, while we have chosen a time step of 0.1 fs and simulated over 2–3 millions steps to ensure relaxation of the spin structures. We employed a semiimplicit scheme proposed by Mentink et al.^{61} to accomplish a time integration of Eq. (13).
Geodesic nudged elastic band method
We calculate the annihilation energy barrier of isolated skyrmions and anitiskyrmions and their collapse mechanism via the geodesic nudged elastic band method (GNEB)^{33,62}. The objective of GNEB is to find a minimum energy path (MEP) connecting initial state (IS), in this case, skyrmion or antiskyrmion, and final state (FS), i.e., FM state, on an energy surface. The GNEB is a chainofstate method, in which a string of images (spin configurations of the system) is used to discretize the MEP. The method selects an initial path connecting IS and FS and systematically brings it to MEP by relaxing the intermediate images. The image relaxation is performed through a force projection scheme, in which the effective field acts perpendicular and spring force acts along the path. The maximum energy on the MEP corresponds to a saddle point (SP) which defines energy barrier separating IS and FS. The energy of the SP is accurately determined using a climbing image scheme on top of GNEB.
Data availability
The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information files. Source data are provided with this paper.
Code availability
The atomistic spin dynamics code is available from the authors upon reasonable request.
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Acknowledgements
We gratefully acknowledge computing time at the supercomputer of the NorthGerman Supercomputing Alliance (HLRN) and financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Project No. 418425860 (Grant No. HE3292/131). We thank Pavel F. Bessarab for valuable discussions.
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S.He. devised the project. S.P. implemented the threesite four spin interaction into the atomistic spin dynamics code and tested the code including HOI. S.Ha., and S.P. performed the DFT calculations. S.Ha., S.P., and S.v.M. performed the spin dynamics and GNEB simulations. S.Ha. and S.P. prepared the figures. S.P. and S.He. wrote the paper. All authors discussed the data and contributed to preparing the paper.
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Paul, S., Haldar, S., von Malottki, S. et al. Role of higherorder exchange interactions for skyrmion stability. Nat Commun 11, 4756 (2020). https://doi.org/10.1038/s4146702018473x
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DOI: https://doi.org/10.1038/s4146702018473x
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