Abstract
Magnetic skyrmion is a topologically protected particlelike object in magnetic materials, appearing as a nanometric swirling spin texture. The size and shape of skyrmion particles can be flexibly controlled by external stimuli, which suggests unique features of their crystallization and lattice transformation process. Here, we investigated the detailed mechanism of structural transition of skyrmion lattice (SkL) in a prototype chiral cubic magnet Cu_{2}OSeO_{3}, by combining resonant soft Xray scattering (RSXS) experiment and micromagnetic simulation. This compound is found to undergo a triangulartosquare lattice transformation of metastable skyrmions by sweeping magnetic field (B). Our simulation suggests that the symmetry change of metastable SkL is mainly triggered by the Binduced modification of skyrmion core diameter and associated energy cost at the skyrmionskyrmion interface region. Such internal deformation of skyrmion particle has further been confirmed by probing the higher harmonics in the RSXS pattern. These results demonstrate that the size/shape degree of freedom of skyrmion particle is an important factor to determine their stable lattice form, revealing the exotic manner of phase transition process for topological soliton ensembles in the nonequilibrium condition.
Introduction
Recently, the concept of topology attracts attention as a source of rich emergent phenomena in condensed matters. In the systems with tensor fields, singular defects called topological solitons often appear, which cannot be erased by the continuous deformation and therefore behave as stable objects^{1}. One typical example is a skyrmion in magnetic materials^{2,3,4,5,6}, which appears as a nanometric swirling spin texture with particlelike character as shown in Fig. 1a. Skyrmion spin texture is generally characterized by nonzero integer skyrmion number N_{sk} described by^{6}
which represents how many times the spin directions wrap a unit sphere. Here, the integral is taken over the twodimensional magnetic unit cell and \({\mathbf{n}}\left( {\mathbf{r}} \right) = {\mathbf{m}}\left( {\mathbf{r}} \right)/{\mathbf{m}}\left( {\mathbf{r}} \right)\) and n_{sk} (r) represent the unit vector pointing in the local magnetization (m(r)) direction and topological charge density, respectively. Interestingly, similar topological solitons are also known to appear in various physical context, such as skyrmions, hopfions, and heliknotons in liquid crystals^{7,8,9,10} or Abrikosov vortices in typeII superconductors^{11}. These particlelike objects generally prefer to form a periodic lattice in the similar manner as atomic or molecular crystals, implying that topological solitons can be a unique building block for a rich variety of tunable ordered structures.
Experimentally, magnetic skyrmions are found in a series of noncentrosymmetric systems, such as metallic B20 (MnSi, FeGe, Fe_{1−x}Co_{x}Si, etc.)^{3,4,6}, CoZnMn alloys^{12}, and insulating Cu_{2}OSeO_{3}^{13,14}. These compounds are characterized by the chiral cubic crystal structure, where Dzyaloshinskii–Moriya (DM) interaction plays a key role in the stabilization of the skyrmion spin texture. In such systems, skyrmions usually crystallize into a closepacked triangularlattice form in the equilibrium condition (Fig. 1b). On the other hand, recent studies on MnSi^{15,16} and CoZnMn alloys^{17,18} have revealed the reversible transition between the triangular skyrmion lattice (SkL) (Fig. 1b) and square SkL (Fig. 1c) as a function of temperature (T) and/or external magnetic field (B) in the nonequilibrium condition. For such a structural transition of SkL, the possible relevance of the magnetic anisotropy has been discussed, while the detailed mechanism is yet to be clarified.
In the present study, we have investigated the microscopic origin of such a symmetry change of magnetic SkL. By performing the smallangle resonant soft Xray scattering (RSXS) experiments for a prototype chirallattice insulator Cu_{2}OSeO_{3}, Binduced triangulartosquare lattice transformation of metastable skyrmions is confirmed. Our micromagnetic simulation, without including magnetic anisotropy term, reveals that the observed SkL transformation is mainly triggered by Bdependent change of skyrmion core diameter and associated energy cost at the skyrmion–skyrmion interface region. Such internal deformation of skyrmion particle has indeed been detected experimentally by measuring the higher harmonics in the RSXS patterns. Our results reveal the unique manner of phase transition process of SkL in the nonequilibrium condition and suggest that the size/shape degree of freedom of skyrmion particle plays an important role in the determination of their stable lattice form.
Results
Our target material Cu_{2}OSeO_{3} is an insulator characterized by a chiral cubic crystal structure with the space group P2_{1}3^{13,14}. The magnetism is governed by the Cu^{2+} ion with S = 1/2 and the DM interaction leads to the longperiod modulation of local magnetic moment direction. To investigate the detailed spin texture, RSXS experiments have been performed in the smallangle scattering geometry for a (001)oriented Cu_{2}OSeO_{3} thin plate (Fig. 1f). Here, the directions of incident soft Xray beam and external magnetic field are fixed perpendicular to the sample surface (i.e., parallel to the outofplane [001] direction). When the Fourier transform of magnetic structure contains the modulated spin component \(({\hat{\mathbf{m}}}({\mathbf{Q}})\exp [i{\mathbf{Q}} \cdot {\mathbf{r}}] + c.c.)\) with \({\hat{\mathbf{m}}}({\mathbf{Q}})\) being a complex vector, the corresponding magnetic scattering intensity I(Q) can be described as \(I({\mathbf{Q}}) \propto \left( {{\mathbf{e}}_i \times {\mathbf{e}}_f} \right) \cdot {\hat{\mathbf{m}}}({\mathbf{Q}})^2\), where e_{i} and e_{f} represent the polarization vectors of incident and scattered beams, respectively^{19}. As the scattered beam is approximately parallel to the incident beam in the smallangle scattering geometry, the component of \({\hat{\mathbf{m}}}({\mathbf{Q}})\) parallel to the outofplane [001] direction (i.e., \(\hat m_z({\mathbf{Q}})\)) is mainly detected in the present measurements.
Figure 2c indicates the B–T magnetic phase diagram for the present sample in the equilibrium condition, determined by the RSXS measurement in the fieldsweeping process after a zerofield cooling (ZFC) (see Supplementary Note I for the detail). Cu_{2}OSeO_{3} hosts helical magnetic order (Fig. 1e) at B = 0, where the neighboring spins rotate within a plane normal to the magnetic modulation vector Q  < 100 > . The application of a magnetic field along the outofplane [001] direction induces the transition into the conical magnetic phase (Fig. 1d), in which the Qvector is aligned parallel to the magnetic field (i.e., Q  B). The equilibrium triangular SkL phase (Fig. 1b) appears in the narrow B–T region just below the magnetic ordering temperature T_{c}, where the spin texture can be approximately described as \({\mathbf{m}}({\mathbf{r}}) = \left( {0,0,m_0^z} \right) + \mathop {\sum}\nolimits_{\upsilon = 1,2,3} {({\hat{\mathbf{m}}}({\mathbf{Q}}_{\upsilon})\exp \left[ {i{\mathbf{Q}}_\upsilon \cdot {\mathbf{r}}} \right] + c.c)}\), with \(m_0^z\) being the Binduced outofplane uniform magnetization component. In this situation, the three Qvectors lie perpendicular to the magnetic field (i.e., Q ⊥ B). The observed B–T phase diagram in Fig. 2c is consistent with the previous reports^{13,14}. It is noteworthy that the lowtemperature disordered skyrmion phase just below B = B_{c} (the magneticfield value required to obtain the saturated uniform ferromagnetic state) reported in the bulk sample^{20,21} is absent in the present thinplate sample.
Next, we investigate the magneticfield variation of the quenched metastable SkL state. By performing a field cooling (FC) passing through the equilibrium SkL phase (black line in Fig. 2a: Path 1), the SkL phase can survive down to lower temperatures as a metastable state. Here, the nucleation probability generally scales with the sample volume and thus the firstorder transition from the SkL phase to the competing conical phase accompanied by the change of topological number can be effectively avoided even with the relatively slow cooling ratio (10 K/min) in such a microfabricated crystal^{22}. In Fig. 2e, the RSXS diffraction pattern obtained at 20 K just after the FC process at +60 mT (Path 1) is indicated. In the present setup, a magnetic field is applied along the outofplane [001] direction and the magnetic modulation vectors within the (001) plane is detected. The observed sixspot diffraction pattern (Fig. 2e) indicates the existence of three Qvectors within a plane perpendicular to B, demonstrating the realization of the metastable triangular SkL. In the Bincreasing process (Fig. 2d), the metastable triangular SkL state with the sixspot diffraction pattern remains up to 150 mT, i.e., just before entering the conical phase, where the Qvector is aligned parallel to the outofplane Bdirection and diffraction spots disappear. In the Bdecreasing process, the triangular SkL state survives even down to negative fields, whereas the sixspot pattern suddenly transforms into a fourspot pattern at −40 mT (Fig. 2f) and then the transition into the conical state takes place at −45 mT (Fig. 2g). The B–T phase diagram for the abovementioned fieldsweeping process (Path 1) is summarized in Fig. 2a. In another fieldsweeping process (black line in Fig. 2b: Path 2), where the magnetic field is returned back to the positive direction after the fourspot pattern once appeared at −40 mT, this fourspot pattern survives up to positive fields (Fig. 2j) and then the sixspot pattern is retrieved above 50 mT (Fig. 2i). From such a reversible change between the fourspot and sixspot patterns, the observed fourspot pattern is assigned to a square SkL state, endowed with the topological charge. Here, a multidomain state of helical magnetic phase (i.e., coexistence of helical domains with Q  [100] and Q  [010]) with zero topological charge can be ruled out as the origin of the fourspot diffraction pattern, as the metastable triangular SkL state cannot be generated by sweeping the field from the equilibrium helical spin state at this temperature. For comparison, the field variation of the diffraction patterns at 20 K after ZFC is shown in Fig. 2k–m, where the helical state with twospot pattern (Q  [100]) at zero field directly changes into the conical state (Q  B) in the fieldincreasing process and no trace of SkL state (\({\mathbf{Q}} \bot {\mathbf{B}}\)) is observed (Fig. 2c). (The results for the different fieldsweeping path and the detailed analysis of the RSXS data are provided in Supplementary Notes II and III).
To investigate the microscopic origin of lattice transformation of SkL, we have performed micromagnetic simulations based on Landau–Lifshitz–Gilbert (LLG) equation by using MuMax3 software^{23}. The stable magnetization distribution m(r) was deduced by minimizing the magnetic free energy \(E = {\int} {\varepsilon \;d{\mathbf{r}}}\) with energy density ε(r) given by
where the first, second, third, and fourth terms represent Heisenberg exchange, DM, Zeeman, and magnetostatic energy, respectively. J, D, and M_{s} represent the magnitudes of exchange interaction, DM interaction, and local magnetic moment, respectively. B_{ext} and B_{d} are the external magnetic field and the demagnetizing field, respectively. A small amount of impurity sites are randomly introduced in the present model (see “Methods” for the detail).
Figure 3 summarizes the realspace distribution of local magnetic moment m (Fig. 3a–d), topological charge density n_{sk} (Fig. 3e–h), and energy density ε (Fig. 3i–l) calculated with various amplitudes of magnetic field based on the micromagnetic simulations. Here, the triangular SkL state is initially prepared at B/B_{c} = +0.44 (Fig. 3a) and then the magnetic field is altered at zero temperature (Fig. 3b–d). At B/B_{c} = +0.44 (Fig. 3a), neighboring skyrmion cores (i.e., regions with negative outofplane moment m_{z}, whose diameter is defined as d) are well separated and negative sign of n_{sk} is concentrated at the center of the core regions (Fig. 3e). For such a positive value of B, the skyrmion core region has higher energy density (Fig. 3i), reflecting the energy cost due to the Zeeman term. With decreasing the magnetic field, the skyrmion core region gradually expands and the intervening region between neighboring skyrmions (i.e., the region with positive m_{z}) squeezes reflecting the modification of the Zeeman energy gain, while the total number of skyrmion cores and their triangularlattice form are still kept unchanged at zero field (Fig. 3b) and even in the negative B (Fig. 3c). By further decreasing B, the transition into the square SkL state happens around B/B_{c} = −0.3, in accord with the experimental observation (Fig. 3d). Importantly, in case of the negative sign of B, the energy cost becomes largest at the interface region between two neighboring skyrmion cores (Fig. 3j–l). It is mainly because this region is characterized by (1) the positive sign of m_{z} with Zeeman energy cost, and (2) the steep spatial change of local moment direction, which disturbs the ideal spin modulation pitch determined by the balance between J and D and hence causes a large exchange energy cost. In particular, the latter contribution becomes more significant as the skyrmion core diameter d becomes larger (i.e., the intervening region becomes narrower), which demands the reduction of the number of energycosting skyrmion–skyrmion interface. As the individual skyrmion particle is surrounded by six (four) skyrmions in the triangular (square) SkL, the energy cost at such interface regions between nearestneighbor skyrmions can be reduced by the transition from triangular SkL into square SkL. Therefore, we can understand that the observed triangulartosquare transformation of SkL is mainly caused by the Bdependent modification of skyrmion core diameter d (In the abovementioned process, spin vortices and antivortices emerge at the intervening region between original skyrmion cores (Fig. 3f–h), whereas their contribution to N_{sk} cancels out and the total topological charge remains unchanged. See Supplementary Note VII).
In principle, such a deformation of skyrmion particle can be evaluated by measuring the relative amplitude of the higher harmonics in the magnetic modulation. Figure 4d indicates the magneticfield dependence of \(\hat m_z(2Q)/\hat m_z(1Q)^2\) calculated from the result of the micromagnetic simulation, with \(\hat m_z(1Q)\) and \(\hat m_z(2Q)\) representing the amplitude of fundamental (1Q) and secondorder harmonic (2Q) modulation component of m_{z}, respectively. As the Bvalue decreases in the triangular SkL state, the skyrmion core diameter d monotonically increases, whereas the coretocore distance a remains constant to keep the total topological number unchanged. At B/B_{c} ~ +0.25, the second harmonic component \(({\mathrm{i}}.{\mathrm{e}}.\hat m_z(2Q)/\hat m_z(1Q)^2)\) is minimized, where d is the half of a and the magnetic modulation is almost sinusoidal. When B becomes larger or smaller than this value, d/a deviates from 1/2 and the larger amplitude of second harmonics is induced. With decreasing B, the second harmonic component in the triangular SkL state reaches a maximum just before the transition into the square SkL state around B/B_{c} = −0.3. The square SkL is stable only for a narrow B range and the further decrease of B leads to the destruction of skyrmions.
To experimentally confirm such Binduced change of skyrmion core size, we investigate the second harmonic magnetic reflections in the RSXS results. It is noteworthy that the RSXS intensity mainly reflects the component of \({\hat{\mathbf{m}}}({\mathbf{Q}})\) parallel to the outofplane [001] direction (i.e., \(\hat m_z({\mathbf{Q}})\)), as discussed previously. In Fig. 4a, the diffraction pattern measured at 0 mT in the metastable triangular SkL state is indicated, where magnetic reflections corresponding to the 2Q modulation components are clearly discerned in addition to the 1Q ones. Figure 4b indicates the linescan profile of diffraction intensities for the 1Q and 2Q reflections measured at 20 K with various amplitudes of B, and the magneticfield dependence of I(2Q)/I(1Q), i.e., the integrated intensity of 2Q magnetic reflection normalized by the 1Q one, is summarized in Fig. 4c. I(2Q)/I(1Q) in the triangular SkL state exhibit the minimum at around +90 mT and a deviation of Bvalues from this minimum leads to the enhancement of second harmonic intensity. As B is further reduced, I(2Q)/I(1Q) ratio monotonously increases and reaches a maximum just before the transition into the square SkL phase at −40 mT. These experimental behaviors are in good agreement with the aforementioned prediction by the micromagnetic simulations (Fig. 4d), which suggests that the observed transition between the triangular and square SkL phases is indeed triggered by the Binduced modification of skyrmion core diameter (see Supplementary Note III for the detailed discussion on the I(2Q)/I(1Q) profile).
Discussion
Previously, the stability of square SkL phase has been discussed in several theoretical works^{2,20,24,25,26}, while they mostly focused on the case of the equilibrium ground state in the positive field region, i.e., not the nonequilibrium metastable state in the negative field region as studied here. As the simplest magnetic Hamiltonian in Eq. (2) generally favors the closepacked triangular SkL phase as the ground state, the additional contribution of magnetic anisotropy and/or higherorder fourspin interaction is required to stabilize the square SkL state in the equilibrium condition according to the previous reports^{20,24,25,26}. On the other hand, our present calculation suggests that these interactions are not necessary to obtain the metastable square SkL state in the nonequilibrium condition, only if skyrmion particles can survive up to a sufficiently large amplitude of negative field during the Bsweeping process. The overall good agreement between the theoretical and experimental results (i.e., the nonequilibrium phase diagram and Bdependence of skyrmion core diameter as shown in Fig. 4c, d) supports the validity of this picture, demonstrating that the Bdependent modification of skyrmion core diameter and associated energy cost at the skyrmion–skyrmion interface region are the key for the observed triangulartosquare SkL transformation. It is noteworthy that for the present case with insulating Cu_{2}OSeO_{3}, the contribution of fourspin interaction (that is generally mediated by itinerant electrons) is negligible, whereas the magnetic anisotropy may also cooperatively promote the appearance of metastable square SkL on the (001) plane. The latter contribution affects the critical Bvalue for the triangulartosquare SkL transition (see Supplementray Note IV. The discussion on the temperature dependence is also provided in Supplementary Note VI).
In the present work, we have clarified the detailed mechanism of magneticfieldinduced structural change of metastable SkL. The observed phase transition process is unique, since it is accompanied by the significant Binduced size change of skyrmion particle, in contrast with the case of conventional atomic or molecular crystals. Interestingly, a similar fieldinduced size/shape change has also been reported for other topological solitons such as heliknotons in liquid crystal systems recently^{10}, where the electricfieldinduced lattice symmetry change and giant electrostriction are observed. Our present results suggest that topological solitons characterized by the flexible particleshape/size degree of freedom commonly host unique manner of crystallization and fieldinduced structural phase transition, which may promise emergence of intriguing phenomena and functions from topologicalparticle ensembles.
Methods
Sample preparation
A single crystal of Cu_{2}OSeO_{3} was grown by a chemical vapor transport method. A (001)oriented thin plate of Cu_{2}OSeO_{3} with about 800 nm thickness was prepared using the focused ion beam (FIB) microfabrication technique (Supplementary Fig. 6). To block the transmission beam, the back side of the Si_{3}N_{4} membrane window was covered with gold film, and subsequently, a pinhole of about 6 μm in diameter was drilled. The sample was mounted to cover the pinhole and attached to the membrane with single tungsten contact to avoid tensile strain.
Smallangle RSXS
Smallangle RSXS experiments were performed using circularly polarized Xray with the resonance energy of 931 eV (i.e., Cu L_{3} absorption edge) in the transmission geometry at the beamline BL16A, Photon Factory, KEK, Japan^{27}. The diffraction patterns were recorded by a directdetection CCD detector, which was protected from the transmitted direct beam with a beam catcher. The magnetic field was applied parallel to the incident Xray beam and perpendicular to the thin plate ( [001]) by a Helmholtz coil. We note that the orientation of the SkL is stochastic and sometimes accompanied by a multidomain SkL state as previously reported in soft Xray scattering experiments in the thinplate samples^{22} and bulk samples^{28,29}.
Micromagnetic simulation
The micromagnetic simulations based on LLG equation were performed with varying bias field by using MuMax3 software^{23}. The magnetization distribution for each value of the bias field was deduced by minimizing the magnetic free energy in Eq. (2). Here we used the material parameters, J = 8.78 × 10^{−12} Jm^{−1}, D = 1.58 × 10^{−3} Jm^{−2}, M_{s} = 3.84 × 10^{5} Am^{−1}, and Gilbert damping constant α = 0.1, reported for FeGe^{30}, which hosts similar magnetic modulation period as Cu_{2}OSeO_{3}. In this case, the magneticfield value required to obtain the saturated uniform ferromagnetic state B_{c} = 1.14 T. The simulation program was run for a system with a size of 1024 × 1024 × 2 nm^{3} modeled with mesh sizes of 2 × 2 × 2 nm^{3} under periodic boundary conditions along the x and y axes. To represent impurities and/or defects due to Ga ion irradiation arising from FIB fabrication process in the present sample, we introduced the easyplane magnetic anisotropy K = 1 × 10^{6} Jm^{−3} at randomly selected sites and set the density of the random impurities to 0.014%. Such defect sites are known to enhance the metastability of SkL state by preventing the firstorder phase transition into the thermodynamically stable nontopological magnetic state^{31}. The initial magnetization distribution at B/B_{c} = 0.44 (Fig. 3a) was prepared by relaxing a spin configuration of a triangular SkL constructed by using the builtin function of MuMax3. The simulation results for the thicker sample is provided in Supplementary Note V.
Data availability
The data presented in the current study are available from the corresponding authors on reasonable request.
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Acknowledgements
We thank X.Z. Yu and M. Mochizuki for helpful discussions, and K. Amemiya and K. Ono for experiment supports. This work was partly supported by JSPS GrantsInAid for Scientific Research (Grants numbers 16H05990, 18H03685, 19H04399, and 20H00349), PRESTO (Grant numbers JPMJPR177A and JPMJPR18L5) and CREST (Grant number JPMJCR1874) from JST, MEXT Quantum Leap Flagship Program (MEXT QLEAP) (Grant number JPMXS0120184122), Research Foundation for OptoScience and Technology, Asahi Glass Foundation, and Murata Science Foundation. Soft Xray scattering work was performed under the approval of the Photon Factory Program Advisory Committee (Proposal numbers 2015S2007 and 2018S2006). V.U. acknowledges support from the SNF Sinergia CRSII5171003 NanoSkyrmionics.
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R.T., S.S., and Y.T. conceived the project. R.T. and S.S. contributed to the sample preparation. R.T., Y. Yamasaki, V.U., Y. Yokoyama, and H.N. performed RSXS measurements and analyzed the data. T.Y. performed micromagnetic simulations. R.T. and S.S. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Takagi, R., Yamasaki, Y., Yokouchi, T. et al. Particlesize dependent structural transformation of skyrmion lattice. Nat Commun 11, 5685 (2020). https://doi.org/10.1038/s41467020194808
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DOI: https://doi.org/10.1038/s41467020194808
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