Abstract
Manipulating topological spin textures is a key for exploring unprecedented emergent electromagnetic phenomena. Whereas switching control of magnetic skyrmions, e.g., the transitions between a skyrmionlattice phase and conventional magnetic orders, is intensively studied towards development of future memory device concepts, transitions among spin textures with different topological orders remain largely unexplored. Here we develop a series of chiral magnets MnSi_{1−x}Ge_{x}, serving as a platform for transitions among skyrmion and hedgehoglattice states. By neutron scattering, Lorentz transmission electron microscopy and highfield transport measurements, we observe three different topological spin textures with variation of the lattice constant controlled by Si/Ge substitution: twodimensional skyrmion lattice in x = 0–0.25 and two distinct threedimensional hedgehog lattices in x = 0.3–0.6 and x = 0.7–1. The emergence of various topological spin states in the chemicalpressurecontrolled materials suggests a new route for direct manipulation of the spintexture topology by facile mechanical methods.
Introduction
The concept of topology provides a powerful scheme for the classification of electronic and magnetic states, and also for the description of their physical properties^{1}. Topology of a magnetic structure is characterized by the winding number \(w = \frac{1}{{8{\mathrm{\pi }}}}\epsilon ^{ijk}{\kern 1pt} {\int}_S {{\kern 1pt} dS_k{\mathbf{n}}({\mathbf{r}}) \cdot [\partial _i{\mathbf{n}}({\mathbf{r}}) \times \partial _j{\mathbf{n}}({\mathbf{r}})]}\). This quantity counts how many times the direction of the local magnetization, i.e., n(r) = m(r)/m(r), wraps the unit sphere within the unit area S. When a magnetic structure possesses a nonzero integer winding number, it behaves as a topologically stable spinobject, producing emergent phenomena unique to its topological class.
A representative example is the magnetic skyrmion, which is a twodimensional (2D) nanometric vortexlike structure consisting of many electron spins^{2,3,4}. In the bulky compounds, skyrmions elongate in cylindrical forms, usually assembling in a hexagonal lattice, i.e., the skyrmion lattice (SkL). In reciprocal space, the hexagonal SkL can be approximately described as a superposition of three helical modulations with the wavevectors (qvectors) forming a mutual angle of 120° in a plane perpendicular to external magnetic field H, as shown in Fig. 1a. Skyrmionhosting materials are of great variety, including noncentrosymmetric bulk magnets^{5} and multilayered thin films^{6,7}. Through the interaction with conduction electrons, skyrmions generate the effective magnetic field confined in each interior, socalled emergent magnetic field as defined by the Berry curvature \(b_k = \frac{1}{2}{\it{\epsilon }}^{ijk}{\mathbf{n}}\left( {\mathbf{r}} \right) \cdot \left[ {\partial _i{\mathbf{n}}\left( {\mathbf{r}} \right) \times \partial _j{\mathbf{n}}\left( {\mathbf{r}} \right)} \right]\). Owing to their topology, skyrmions carry the quantized emergent flux \(\phi _0 =  \frac{h}{e}\) in case of the strong spincharge coupling, which offers attractive spintronic functionalities conserved even in nanoscale devices;^{8,9,10} such as topological Hall effects^{11,12,13} and emergent electromagnetic inductions^{1,14,15,16}.
Also as a result of the geometric constraint, transitions of topological spin textures are accompanied by dynamics of topological spin defects, as exemplified by the emergence of spin hedgehogs in the course of creation or annihilation of skyrmions^{17,18}. Because transformations between skyrmions and conventional magnetic orders need discrete changes in the winding number, they cannot be realized by smooth change of the directions of local spins. Instead, it is necessary to introduce hedgehog point defects, which are threedimensional (3D) topological spin structures, behaving as emergent magnetic monopoles or antimonopoles with nonzero divergence of emergent magnetic field \(\left( {\frac{1}{{4{\mathrm{\pi }}}}\nabla \cdot {\mathbf{b}} = \pm 1} \right)\)^{19}. The motions of hedgehogs locally give or remove the topological winding number, causing the topological transition through the elongation, contraction, coalescence, and division of skyrmion strings^{17,18}. In association with such dynamics of topological charges, the topological transition of a spin texture often involves nontrivial emergent phenomena, e.g., the formation of fluctuating topological magnetic order^{20}, the concomitant nonFermiliquid like behavior^{21} and electrical magnetochiral effect^{22}. Spin hedgehogs are also found as a dense lattice form in MnGe, namely the array of hedgehogs and antihedgehogs connected by skyrmion strings. This state can be described approximately by three helical modulations with their qvectors forming the orthogonal sides of a cube^{23} and is here referred to as cubic3q hedgehog lattice (HL) (Fig. 1c). The transition from the HL state to the nontopological state, e.g., singleq conical and ferromangnetic state, undergoes the pair annihilation of hedgehogs and antihedgehogs^{24}, which entails critical anomalies of resistivity, elastic property, and thermopower as well^{24,25}.
In order to harness the topological properties unique to each spin texture and to explore nontrivial emergent phenomena at their transitions, direct control of topology of spin texture is essential. In this context, switching of the spin textures among plural different topologically nontrivial classes has remained a challenge. We focus on chemical/mechanical pressure as one potential approach to this end, by achieving dramatic modification in magnetic interactions through changing interatomic distances.
Here we report on the topological transitions among 2D SkL and 3D HLs in cubic chiral magnets MnSi_{1−x}Ge_{x}. By changing chemical pressure through substitution between Si and Ge, in other words, by controlling the lattice constant a (Supplementary Fig. 1), we observed that the SkL in MnSi undergoes twostep transitions to the cubic3q HL in MnGe. In the intermediate composition range, we unveiled a new topological spin texture characterized by four qvectors pointing in the apical directions of a regular tetrahedron. This state corresponds to the facecenteredcubic array of the hedgehogs and antihedgehogs, which we call tetrahedral4q HL (Fig. 1b). Our neutron scattering experiment and Lorentz transmission electron microscopy (LTEM) observation confirm the conventional SkL for Sirich composition range (x = 0–0.25), tetrahedral4q HL for the intermediate range (x = 0.3–0.6), and cubic3q HL for Gerich range (x = 0.7–1). Furthermore, by highfield Hall resistivity measurements, we identified topological Hall effect of Berryphase origin in each magnetic phase, supporting their topological spin arrangements.
Results
Overview of topological magnetic transitions in MnSi_{1−x}Ge_{x}
To overview the magnetic transitions in MnSi_{1−x}Ge_{x}, we first show the magnetic phase diagrams for varying x as summarized in Fig. 1d. (See Supplementary Fig. 2 for magnetic transition temperatures T_{N} for varying x in detail.) The drawn phase boundary line for each x represent the ferromagnetic transition, determined from magnetization measurements. (See Supplementary Fig. 3 for details.) The variation in the phase diagram indicates that MnSi_{1−x}Ge_{x} can be sorted into three categories, as evident from the distinct value of ferromagnetictransition magnetic field (H_{c}) at the lowest temperature; two steep changes in H_{c} at x ~ 0.3 and x ~ 0.7 (ΔH_{c} ~ 10 T) suggests the magneticstructure transitions (Fig. 1e). There are also recognized characteristic profiles in xdependence of other magnetic properties. As for the saturation magnetization (M_{s}), a plateaulike structure appears around M_{s} ~ 1 μ_{B}/f.u. within the range x = 0.4–0.6 (Fig. 1f). Incidentally, the similar feature was observed in a former study on hydrostaticpressure dependence of M_{s} in MnGe^{26}.
SANS and LTEM studies at zero magnetic field
In order to clarify the variation of magnetic period (λ) and direction of wavevectors (qvectors) in MnSi_{1−x}Ge_{x}, small and wideangle neutron scattering experiments as well as LTEM observations were performed at zero magnetic field. Figure 2a–c show the SANS intensity patterns of x = 0.2, 0.6, and 0.8 after zero field cooling from room temperature. The observed Debyeringlike patterns indicate the formation of periodically modulated magnetic structures. Their modulation periods are determined from the radius (q) of the each diffracted ring pattern as λ = 2π/q (Fig. 2h).
The qvectors, which are fixed along specific crystalaxes due to magnetic anisotropy, were identified by using LTEM for x = 0.2 (Fig. 2d, e) and by wideangle neutron scattering for x = 0.4, 0.6, and 0.8 (Fig. 2f, g). (See also Supplementary Fig. 6 for all the data sets.) In a thin plate sample of x = 0.2, there observed a helical structure with q  〈100〉 and λ = 9 nm (Fig. 2e). Here we note that the observed qdirection is not necessarily the same in bulk samples as seen in the LTEM study on MnSi, where the helical direction is dependent on the crystalline orientation due to large magnetic anisotropy effects at the surfaces^{27}. Since the qdirections in bulks of MnSi and x = 0.4 (Supplementary Fig. 6b) are both parallel to 〈111〉, we speculate that it is also the same in the x = 0.2 bulk sample, although the direct verification by wideangle neutron scattering is difficult within the current qresolution. (See Supplementary Fig. 6a). In powder samples of x = 0.6 and 0.8, we show the wideangle neutron diffraction profiles around (110) nuclear reflection in Fig. 2f, g. Because magnetic reflections generally appear as satellite peaks at the wavenumber of \(\left {{\mathbf{q}}_{\mathrm{n}} \pm {\mathbf{q}}} \right\) in a powder neutron diffraction, we can determine the qdirection from the satellite peak position once we identify the magnitude of q by SANS^{13}. Here q_{n} is the reciprocal lattice vector. Each satellite peak observed in x = 0.6 and 0.8 (indicated by blue and red triangles in Fig. 2f, g) can be indexed as \(\left( {\frac{{2{\mathrm{\pi }}}}{a}  \frac{q}{{\sqrt 3 }},\frac{{2{\mathrm{\pi }}}}{a}  \frac{q}{{\sqrt 3 }}, \pm \frac{q}{{\sqrt 3 }}} \right)\) and \(\left( {\frac{{2{\mathrm{\pi }}}}{a}  q,\frac{{2{\mathrm{\pi }}}}{a},0} \right)\), respectively. Namely, the qvector directions are along 〈111〉 crystal axes for x = 0.6, and 〈100〉 crystal axes for x = 0.8.
As summarized in Fig. 2h, there are two features in xdependence of the magnetic modulation q; the twostep magnetic transitions observed as distinct changes in the phase diagram (Fig. 1d). One is the discontinuous variation in λ (=2π/q) at x ~ 0.3 and the other is change in the pinned qdirection at x ~ 0.7. It is therefore clear that three distinct magnetic phases are realized in MnSi_{1−x}Ge_{x}. In later sections, those respectively turn out to be helicl/SkL (x = 0–0.25), tetrahedral4q HL (x = 0.3–0.6), and cubic3q HL (x = 0.7–1).
Formation mechanisms of shortperiod topological spin textures
It is noteworthy that such remarkable variations in the magnetic properties can be driven essentially by such a latticeconstant change without any change in lattice symmetry. Here we discuss the origin of the dramatic magnetic transitions in MnSi_{1−x}Ge_{x}. One important indication we obtain is that the dominant magnetic interactions for the 3D HL states may differ from those for 2D SkL where the competition between ferromagnetic exchange interaction (EXI) \(( { \propto J{\mathbf{S}}_i \cdot {\mathbf{S}}_j})\) and DzyaloshinskiiMoriya interaction (DMI) \(( { \propto {\mathbf{D}} \cdot ({\mathbf{S}}_i \times {\mathbf{S}}_j)})\) determines the basic magnetic properties^{28}, such as \(\lambda \sim a \cdot J{\mathrm{/}}D\) (a being the lattice constant) and \(H_{\mathrm{c}}\,\sim \,D^2M{\mathrm{/}}J\). In the conventional SkL materials, the magnetic ground state is a longperiod helical structure (λ typically ranges from 10 to 100 nm) with relatively small \(H_{\mathrm{c}}\) (<1 T) since the energy scale is wellseparated as \(J \gg D\). If the same model could be applied, the extremely short \(\lambda\) (1.94–2.80 nm) and large \(H_{\mathrm{c}}\) (12–24 T) observed in MnSi_{1−x}Ge_{x} (x = 0.4–1) would require much larger DMI even exceeding EXI. The band structure calculations^{29}, however, demonstrate a contradictory behavior of DMI in MnSi_{1−x}Ge_{x} to this naive expectation: DMI gets rather smaller with increasing x, whereas the calculated values of M_{s} are in accord with the observed ones (Supplementary Fig. 4). Thus, the DMI may not be the primary origin of the shortperiod helical structure in 3D HL. Instead, the magnetic frustration or RudermanKittelKasuyaYosida (RKKY) interaction^{30,31,32,33} causing competing ferromagnetic and antiferromagnetic EXIs can be a possible mechanism. Possibly related to such conductionelectron mediated exchange interactions, we found that the strong HubbardU implemented in the band structure calculation favors a shortperiod helical structure (Supplementary Fig. 5). Further theoretical studies are desired to clarify the crucial magnetic interaction, which takes over from DMI in the course of enlarging the lattice constant.
SANS and LTEM studies under magnetic fields
Having confirmed the variation of magnetic properties in MnSi_{1−x}Ge_{x}, we investigated the magnetic structures under magnetic fields by SANS in x = 0.2, 0.6, and 0.8 (as representative compositions of the three magnetic phases), and by LTEM in the thin plate of x = 0.2. Magnetic field (H) is applied perpendicular to the incident neutron beam for SANS experiment (Fig. 3a), while H is applied parallel to the electron beam for LTEM measurement (Fig. 3b). As for x = 0.2, LTEM directly reveals the formation of a hexagonal SkL in the plane perpendicular to H, which is also confirmed by the sixfold Fourier transform image (Fig. 3c).
In Fig. 3d–f, we display all the measurement points (gray dots) and sequences (blue arrows) of SANS in the magnetic phase diagrams, and the representative SANS patterns obtained at the T–H points (highlighted by blue stars in Fig. 3d–f) are shown in Fig. 3g–i. (See Supplementary Figs 7–9 for the SANS data at other temperatures and magnetic fields). The SANS pattern on the polycrystalline \(x = 0.2\) compound (Fig. 3g) exhibits intensity peaks parallel (\(\varphi = 0^\circ\), \(180^\circ\)) and perpendicular (\(\varphi = \pm 90^\circ\)) to H, which indicates the conical state modulating along H coexists with SkL. Because only a small portion of SkL states in powder grains can meet the diffraction condition as illustrated in Fig. 3j, the much weaker intensity peaks were observed at \(\varphi = \pm 90^\circ\) than those scattered at \(\varphi = 0^\circ\), \(180^\circ\) by the conical state. Incidentally, we also confirmed that the peaks at \(\varphi = \pm 90^\circ\) only emerge in the SkL phase identified by the magnetization measurements. (See Supplementary Fig. 7 for details).
In x = 0.6 and 0.8, we also detect characteristic SANS patterns under H: In addition to the intensity peaks at \(\varphi = 0^\circ\), \(180^\circ\), there appear peaks at \(\varphi = \pm 70^\circ , \pm 110^\circ\) for \(x = 0.6\) (Fig. 3h) and at \(\varphi = \pm 90^\circ\) for \(x = 0.8\) (Fig. 3i). We interpret these SANS patterns in terms of multipleq structures. As for x = 0.6, Fig. 4 illustrates the possible multipleq structure explaining the SANS result, i.e., the tetrahedral4q state. One of the four qvectors flips along the Hdirection, as is the case of many other B20type magnets^{28}, and generates scattering intensity at \(\varphi = 180^\circ\) (\(0^\circ\)), while the remaining three qvectors produce intensities at \(\varphi = \pm 70^\circ\) \((\pm 110^\circ)\) (Fig. 3k). As for x = 0.8, the observed intensity pattern is essentially identical to that of MnGe where the cubic3q HL is realized^{23}. In this case, one of the three qvectors flips along the Hdirection \(\varphi = 180^\circ\) (\(0^\circ\)), while the other two orthogonal qvectors generate scattering intensities at \(\varphi = \pm 90^\circ\), as illustrated in Fig. 3l. Importantly, these 3q and 4q HL states dominates the whole magneticorder phase below H_{c} shown in Fig. 1d, whereas the SkL state in the Sirich region is generated by variation of H from the nearby helical/conical states (see Supplementary Fig. 7) as commonly observed in the SkLhosting chiral magnets^{28}.
Here we note that we cannot exclude a possibility of multidomain state of the singleq helical structure on the basis of the SANS results alone. As demonstrated in the following section, however, such a scenario is incompatible with the observed large topological Hall effect, which is the hallmark of the formation of noncoplanar spin textures endowed with scalar spin chirality.
Highmagneticfield measurements of topological Hall effect
Outcomes of these topological spin arrangements show up in the large topological Hall responses arising from their scalar spin chirality or skyrmion number^{34}. The measured Hall resistivity \(\rho _{yx}\) for \(x = 0.2,\,0.6,\,0.8\) are summarized in Fig. 4a–c. We estimate the contributions from normal (\(\propto H\)) and anomalous (\(\propto M\)) Hall effects^{35} by reproducing \(\rho _{yx}\) in the ferromagnetic (fieldinduced spin collinear) region with use of the fitting curve \(\rho _{yx}^{{\mathrm{fit}}} = R_0H + R_{\mathrm{s}}\rho _{xx}^2M\), where \(R_0\) and \(R_{\mathrm{s}}\) are the normal and anomalous Hall coefficients, respectively (see Supplementary Fig. 3 for \(M\) and Supplementary Fig. 10 for \(\rho _{xx}\)). Topological Hall effect (THE) arises as the deviation from the conventional contributions: \(\rho _{yx}^{\mathrm{T}} = \rho _{yx} \) \(\rho _{yx}^{{\mathrm{fit}}}\). In MnSi_{1−x}Ge_{x} (x = 0.2) hosting the SkL state, overall Hdependence of \(\rho _{yx}\) obeys that of M (Supplementary Fig. 3), that suggests the dominant contribution from the conventional anomalous Hall effect (Fig. 4a). Besides, we identified negative \(\rho _{yx}^{\mathrm{T}}\) in the intermediate Hregion below H_{c} above 12 K (Fig. 4d). As seen in the color map of \(\rho _{yx}^{\mathrm{T}}\) in Fig. 4g, \(\rho _{yx}^{\mathrm{T}}\) reaches its maximum magnitude of \(\left {\rho _{yx}^{\mathrm{T}}} \right = 30\,{\mathrm{n}}\Omega {\mathrm{cm}}\) near the center of SkL phase; and finite \(\rho _{yx}^{\mathrm{T}}\) persists even in the conical phase, i.e., in the Hregion between SkL and ferromagnetic states, implying that a part of skyrmions remain to subsist perhaps as disordered aggregation^{4}.
In contrast, \(\rho _{yx}\) in MnSi_{1−x}Ge_{x} (x = 0.6 and 0.8) with HL states clearly indicate the large deviation from the conventional Mproportional profile, as shown in Fig. 4b, c. The estimated \(\rho _{yx}^T\) shows complex behaviors with sign changes against T and H variations, as well as oneorderofmagnitude larger values than that in x = 0.2 (Fig. 4e, f, h, i.). The observed THE not only corroborate the formation of the topological multipleq spin textures, but also exhibit different behaviors between tetrahedral4q HL and cubic3q HL. To be specific, MnSi_{1−x}Ge_{x} with cubic3q HL (x = 0.7–0.9 and MnGe^{13}) share similarity in sign change of \(\rho _{yx}^{\mathrm{T}}\): negative \(\rho _{yx}^{\mathrm{T}}\) in the lowT and lowH region gives way to positive \(\rho _{yx}^{\mathrm{T}}\) in the highT and highH region (Fig. 4f, i and Supplementary Fig. 11). The negative and positive \(\rho _{yx}^{\mathrm{T}}\) may be attributed to static and fluctuating effects of emergent magnetic field^{13,24,36}. As for x = 0.6 compound with tetrahedral4q HL, the magnitude of \(\rho _{yx}^{\mathrm{T}}\) is gigantic as well, while its T and Hdependences are complicated to interpret (Fig. 4h). We note that such complex \(\rho _{yx}^{\mathrm{T}}\)profiles with sign changes are also reported at the transitions between versatile topological spin structures, including 4qHL, in SrFeO_{3}^{37,38}. By analogy with it, the sign change of \(\rho _{yx}^{\mathrm{T}}\) at low temperatures in x = 0.6 (e.g., T = 10 K in Fig. 4e) may indicate a transition into different multipleq states or the fieldinduced modification of the 4q structure, which remains an open question. We also note that there may exist robust or pinned excitations of spin hedgehogs even in the nominally ferromagnetic region^{25}, which may imperil the validity of the present estimation of topological Hall effect. Hence, the magnitude and the sign changes of topological Hall resistivity may be difficult to quantitatively elucidate at the moment, while the presence of noncoplanar spin texture manifests itself by such anomalously large signals of topological Hall resistivity.
Discussion
The present results not only on the SANS and LTEM but also on the topological Hall effects unveil the transitions among distinct topological spin textures, namely 2D SkL and two classes of 3D HLs in cubic chiral magnets MnSi_{1−x}Ge_{x}. Compared with the case of SrFeO_{3} where magnetic domains with different helicity and vorticity degrees of freedom should coexist due to the centrosymmetric crystal structure^{38}, a point of uniqueness in MnSi_{1−x}Ge_{x} is ascribed to the fixed helicity and vorticity over the whole chiral lattice. In addition, the transitions between different topological spin textures can be realized simply by controlling lattice constant; therefore, it would be possible to switch the topology of spin textures by application of small pressure or strain, once a composition x is tuned to the transition points (x ~ 0.3 and 0.7). Given that pressure is a fundamental variable that controls the properties of materials by changing interatomic distance or electron transfer interaction^{39}, the impact of pressure on topological spin textures deserves further investigations in a wide range of materials in the light of the exploration of novel spin textures and emergent electrodynamics.
Methods
Sample preparation
Polycrystalline samples of MnSi_{1−x}Ge_{x} were prepared by the highpressure synthesis technique. Mn, Si, and Ge were first mixed with stoichiometric ratio and then melted in an arc furnace under an argon atmosphere. Afterwards, it was heated at 1073 K for 1 h under 5.5–6.0 GPa with a cubicanviltype highpressure apparatus. Powder xray analyses confirmed B20type crystal structure (P2_{1}3).
Magnetic and transport property measurements
Magnetization was measured either by using ACMS option with Physical Property Measurement System (PPMS) or by DC option with Magnetic Property Measurement System (MPMS). Magnetoresistivity and Hallresistivity up to 14 T were measured by using ACtransport option with PPMS. Magnetic field was applied perpendicular to electrical current. Higherfield measurements of magnetization and Hall resistivity were performed utilizing nondestructive pulse magnets energized by capacitor banks and a flywheel DC generator installed at International MegaGauss Science Laboratory of Institute for Solid State Physics (ISSP), University of Tokyo, Japan, respectively. The highfield magnetization was measured up to 56 T by the conventional induction method, using coaxial pickup coils. The highfield resistivity was measured up to 30T using the long (~1s) field pulse with the AC fourwires method employing a numerical phase detection technique with a sampling rate of 200,000 data points per second and an excitation current of 10 kHz and 200 mA_{pp}.
Lorentz TEM observations
Lorentz TEM observations for a (001) MnSi_{0.8}Ge_{0.2} thin plate were performed using a multifunctional transmission electron microscope (JEM2800, JEOL) equipped with doubletilt helium cooling holder (Gatan ULTDT). The thin plate was prepared by an Ar^{+} milling process after mechanical polishing of the bulk sample.
Neutron scattering
Neutron scattering experiments were performed at small and wideangle neutron scattering instrument (TAIKAN) built at BL15 of Materials and Life Science Experimental Facility (MLF) in Japan Proton Accelerator Research Complex (JPARC)^{40}. A powder sample of MnSi_{1−x}Ge_{x} was packed in an aluminum container filled by He gas, and installed in a cryomagnet. The weight of the powder sample was \(0.450\,{\mathrm{g}}\left( {x = 0.2} \right)\), \(0.719\,{\mathrm{g}}\left( {x = 0.4} \right)\), \(1.059\,{\mathrm{g}}\left( {x = 0.6} \right)\), and \(0.751\,{\mathrm{g}}\left( {x = 0.8} \right)\). Magnetic field was applied perpendicular to the incident neutron beam. The diffracted neutron beam with the wavelength of \(0.5 < \lambda < 7.8\) Å was collected by four detector banks of small, middle, and highangle and backward detector banks, and analysed by using timeofflight (TOF) method.
Band structure calculations
Electronic structure calculations were performed using the planewave basis set with the projector augmented wave (PAW) scheme^{41} as implemented in the Vienna Ab initio Simulation Package (VASP)^{42,43}. The PerdewBurkeErnzerhof (PBE) exchangecorrelation functional^{44} and a cutoff energy of 500 eV were used. Experimental crystal structures^{45,46} were adopted and the lattice constants were changed by fixing the fractional coordinates of the atoms. DzyaloshinskiiMoriya interactions were evaluated using a spin current formalism^{29,47}. To calculate the energy of the spiral spin structures, a generalized Bloch theorem was employed^{48}. The effect of strong correlation was discussed using the DFT + U method^{49}.
Data availability
The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank Y. Taguchi and A. Kitaori for fruitful discussions. The neutron experiment at the Materials and Life Science Experimental Facility of JPARC was performed under the user program (Proposal No. 2017L0701 and No. 2016C0002). This work was supported by JSPS KAKENHI (Grants No. 24224009 and No. 18K13497, and No. 17H02815) and JST CREST (Grant No. JPMJCR16F1 and JPMJCR1874).
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Y.F. synthesized polycrystalline samples of MnSi_{1−x}Ge_{x} and performed magnetization and transport measurement by using PPMS/MPMS. Neutron scattering experiment were performed by Y.F., N.K., and T.N. with the support from K.O., Y.K., K.Ka., and T.A. LTEM observation was done by X.Z.Y. Magnetization and Hall resistivity measurement under high magnetic field was performed by Y.F., N.K., H.M., A.M., K.A., and A.M. under the supervision of M.T. and K.Ki. Band structure calculations were performed by T.K. and R.A.; Y.F. and N.K. wrote the draft with the support from Y.T. All the authors discussed the results and commented on the manuscript. Y.T. conceived and organized the project.
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Fujishiro, Y., Kanazawa, N., Nakajima, T. et al. Topological transitions among skyrmion and hedgehoglattice states in cubic chiral magnets. Nat Commun 10, 1059 (2019). https://doi.org/10.1038/s41467019089856
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