Topological stability of spin textures in Si/Co-doped helimagnet FeGe

Element substitutions with magnetic or non-magnetic atoms are known to significantly impact the magnetic structure and related transport properties of magnets. To clarify the change of magnetic structure of B20-type magnets with element doping, we conduct real-space observations of spin textures and their temperature (T)-magnetic field (H) phase diagrams of a helimagnet FeGe with partially substituting Fe and Ge with Co and Si, respectively. The helical period (λ) changes dramatically by the element doping: λ increases by 147% to 103 nm in 30% Co-doped FeGe, whereas it decreases by around 70% to 49 nm in 30% Si-doped FeGe, compared to the λ =70 nm in FeGe. Upon applying the magnetic field normally to (001), (110), and (111) thin plates of both FeSi0.3Ge0.7 and Fe0.7Co0.3Ge, the hexagonal skyrmion crystal (SkX) state emerges. The magnetic phase diagrams observed through the real-space imaging reveal that (1) the SkX can extend to a larger T-H window by reducing the sample thickness or by cooling the sample under specific magnetic fields from temperatures above the transition temperature (TC ); (2) the stability of the SkX phase differs between Si-doped and Co-doped FeGe: the SkX phase is most unstable in the (111) FeSi0.3Ge0.7, while it remains robust in the (111) Fe0.7Co0.3Ge. These differences indicate distinct anisotropic behavior in FeGe with magnetic (Co) and non-magnetic-element (Si) dopants.

Among the B20-helimagnet family, FeGe, exhibiting a higher T C near room temperature (∼280 K) and a relatively short wavelength (λ ∼ 70 nm) [22], is a prototype helimagnet hosting skyrmions.The wavevector Q of the helical structure exhibits temperature dependence in bulk FeGe: Q || ⟨100⟩ at temperatures above 245 K, while it changes to Q || ⟨111⟩ at temperatures below 211 K [23].In thin FeGe plates, however, the Q predominantly aligns along ⟨100⟩ and depends less on temperature [22].Real-space observations have demonstrated that as the sample thickness approaches or falls below λ, the field-induced conical structure could be suppressed, promoting the formation of the skyrmion state.Consequently, the skyrmion crystal (SkX) state extends over a larger region in the temperature-field (T-H) plane [22], in contrast to the small pocket of SkX observed in bulk samples [24].Therefore, controlling the sample's dimensionality is crucial for tuning the state of topological spin textures and their stability.
The exploration of element doping, using both non-magnetic element Si and magnetic element Co in FeGe, has led to intriguing advancements, such as controlled changes in λ, T C , and stability of SkX as well as the corresponding transport properties (anomalous and topological hall resistivity) [25][26][27][28][29][30][31].In Si-doped FeGe, both λ and T C decrease with increasing Si composition [27].In Co-doped FeGe, a transition from helical structures to a ferromagnetic (FM) state occurs when the Co composition x approaches 0.6 due to the competition between cubic anisotropic energy and the DM interaction [28].So far, while the spontaneous formation of vortex (meron)-antivortex (antimeron) pairs are observed as x = 0.5 [32].The doping systematics in topological spin textures remains elusive for x < 0.5.Despite extensive research efforts on profound features of topological spin textures, such as spin spiral modulation in various dimensions [29], anisotropic effects [28], and surface twist instability [30,31], there remains significant interest in understanding the stability of topological spin textures in FeGe when elements are doped into the material.
In this study, we investigate the stability of the SkX state in Si-and Co-doped FeGe thin plates oriented along the (001), (110), and (111) crystal planes using real-space observations with Lorentz transmission electron microscopy (LTEM).The results reveal distinct trends in λ and T C as the doping levels (x for Co and y for Si) increase: in the Co-doped case, λ increases while T C decreases with increasing x; in the Si-doped case, both λ and T C decrease with increasing y.Systematic observations demonstrate that the SkX state remains robust in the Fe 0.3 Co 0.7 Ge (111) plane, while it becomes relatively unstable in the FeSi 0.3 Ge 0.7 (111) plane.Additionally, through the field cooling (FC) process, the metastable SkX phase in Co/Si-doped FeGe thin samples persists at low temperatures (100 K) and under zero magnetic field.Furthermore, reducing the thickness of the thin plate extends the SkX phase over a wider range in the T-H plane.

Experimental method
Single crystals of Si-doped FeGe were grown using the chemical vapor transport method.The crystal growth process utilized iodine as a transport agent with a controlled temperature spanning from 870 • C to 810 • C over a duration of 6 weeks, which was preceded by a pretreatment involving an inverted temperature for 12 h.The Co-doped FeGe polycrystals were synthesized using a high-pressure technique.The Fe, Co, and Ge were mixed stoichiometrically and melted in an arc furnace under an argon atmosphere.The molten mixture was then heated at 1073 K for 1 h under 5 GPa pressure using a cubic-anvil-type high-pressure apparatus.Powder x-ray analyses were conducted to confirm that all the doped samples in this study exhibit B20-type crystal structures.Thin plates of the doped FeGe samples, intended for LTEM observations, were prepared by using the focused ion beam system (Hitachi, NB5000).The crystalline orientations of thin plates were verified by the selected-area electron diffraction.The thickness of each thin plate was roughly determined through a combination of the electron energy-loss spectroscopy [33] and scanning electron microscopy.
LTEM observations were then performed using a transmission electron microscope (JEM-2800, JEOL) operated at a field-free low magnification mode with an accelerating voltage of 200 kV.The external magnetic field was applied normally on the plate plane via tuning the objective magnetic lens current.The magnetic inductions appear as either bright or dark contrasts on the defocused image planes due to the convergence or divergence of the electron beam caused by the Lorentz force.Notably, the image plane displayed contrast reversals between underfocused and overfocused images.To explore the temperature dependence of the magnetic structures, we utilized a liquid nitrogen cooling holder (Gatan, Model 636).The zero-field cooling (ZFC) and FC illustrated in the insets of figures 3(a) and (d) show the cooling processes from room temperature to the target temperature under zero and specific fields, respectively.The in-plane magnetization textures was then extracted from the LTEM images using a commercial software package QPt (HREM Co.), which is based on the transport-of-intensity equation (TIE) [34].

Results and discussion
Figure 1 presents an overview of the magnetic properties resulting from the doping of Si or Co in FeGe.The T C in figure 1(a) is defined as the transition temperature at which the helical structures disappeared during a temperature sweep from 100 K to room temperature.In both Si-and Co-doped FeGe samples, T C decreases as the doping level increases.The decrease in T C of FeSi y Ge 1−y can be caused by the reduction in atomic radius when the Ge is substituted by the Si, which results in a decrease of the density of states around the Fermi level [35].This reduction enhances the electron transfer between transition metal atoms, leading to the broadening of bandwidths and the energy gap, and hence to the decrease of T C [35,36].On the other hand, the T C in Fe 1−x Co x Ge shows a monotonic decrease as the Co-concentration x increases, which is   H, Rq-H, Rq-H + PCSk, SkX, C + ISk, and FM stand for the in-plane helical state, rotating-q helical state, mixed rotating-q helices and partially crystallized skyrmions, SkX state, mixed conical structures and isolated skyrmions, and field-induced ferromagnetic state, respectively.All LTEM images were taken at 140 K with a defocused value of −240 µm.consistent with previous experiments [37,38].Figure 1(b) shows the change in λ (∼J/D) as a function of doping concentration at 100 K, where J represents the exchange interaction strength and D denotes the DM interaction constant.In the case of Co-doped FeGe, λ increases with the doping level x.However, in the case of Si-doped FeGe, λ exhibits an opposite trend, decreasing with the increase in Si composition y.
To explore the impact of crystal orientations on magnetic structures, LTEM observations are performed on both FeSi 0.3 Ge 0.7 (figure 2) and Fe 0.7 Co 0.3 Ge (figure S1) thin plates, normally to the [001], [110], and [111] axes.Figures 2(a)-(c) show the zero-field helical structures at 100 K in (001), ( 110) and (111) planes, respectively.The λ in these thin plates is approximately 49 nm, which is independent of the crystal plane.The white arrows in the images show the in-plane Q-vector directions.In the (001) and (110) thin plates, the Q vectors align along the ⟨010⟩ axes, while in the (111) thin plate it turns to the [ 11 2] or [ 101] axis.The Q vector does not change in a heating run from 100 K to T C .Upon a normal magnetic field, the helices evolve into SkX.Figures 2(d)-( 001) and (110) thin plates, due to the differing signs of crystal chirality in various crystal grains within the polycrystalline sample [39,40].The lattice parameter of SkX (a sk ) are about 56 nm and 119 nm for FeSi 0.3 Ge 0.7 and Fe 0.7 Co 0.3 Ge, respectively, well in accord with the equation λ = √ 3a sk /2.Furthermore, the magnetic-field manipulation of the spin textures is explored.Figure 3 shows a typical variation of magnetic textures in the FeSi 0.3 Ge 0.7 (110) thin plate at 140 K with increasing the magnitude of the normal field.At zero field, the helical propagation vector is oriented along the [001] axis (figure 3(a)).Upon a 72 mT field, a helical state with field induced rotating-q (Rq-H) vector shows up in figure 3(b).Further increase in the applied field to 95 mT induces a partial transition of helical structures into partially crystallized skyrmions (PCSk), creating a mixture of Rq-H structures and PCSk (figure 3(c)).Subsequently, a complete transformation into the SkX state (figure 3(d)) can be observed at a higher field of 226 mT.As the applied field is increased further to 302 mT, the SkX state transforms into a composite configuration of conical structures and isolated skyrmions (C + ISk) (figure 3(e)).Ultimately, with a substantial field enhancement up to 324 mT, a single FM domain state appears where the magnetization aligns along the applied field (figure 3(f)).Next, we investigated the stability of the SkX phase with changing the crystalline orientations, specifically (001), (110), and (111), in both FeSi 0.3 Ge 0.7 and Fe 0.7 Co 0.3 Ge samples.Figure 4 illustrates the magnetic phase diagrams constructed for these orientations from the LTEM analysis of their respective thin plates.Figures 4(a 111) planes, respectively.The thin plates in this case have a thickness of around 120 nm.The density of skyrmions in FeSi 0.3 Ge 0.7 (∼340 skyrmions per square micrometer) is roughly four times larger than that in Fe 0.7 Co 0.3 Ge (∼78 skyrmions per square micrometer).The skyrmion diameter in FeSi 0.3 Ge 0.7 is about two times smaller than that in Fe 0.7 Co 0.3 Ge.In contrast to the unstable SkX in FeSi 0.3 Ge 0.7 (111) plate, the SkX is most stable in the Fe 0.7 Co 0.3 Ge (111) thin plate, similar to that observed in MnSi [41].It has been reported that when the thin plate thickness is greater than 4λ, the SkX is unstable in the MnSi (111) thin plate.However, when the thin plate thickness is less than 2λ, the SkX becomes robust in the MnSi (111) thin plate.Aside from the study of ZFC thermodynamic phase diagrams, we also investigate the meta-stable SkX phase diagram under FC.Figures 4(d) and (h) present the phase diagrams of the FeSi 0.3 Ge 0.7 (110) thin plate and Fe 0.7 Co 0.3 Ge (001) thin plate under FC condition of 146 mT and 50 mT, respectively.In both cases, the SkX phase extends to a larger T-H window, and the SkX is metastable at zero field.Figure S2 illustrates the LTEM images of the (110) metastable SkX state under FC condition, which show no significant transitions to other magnetic structures, even at zero field.
For more comprehensive understanding of the thickness dependence of SkX stability, we extend our investigation to explore the ZFC phase diagrams by varying the sample thickness.Figures 5(a) and (b) illustrate phase diagrams of FeSi 0.3 Ge 0.7 (110) (at 120 nm and 170 nm) and Fe 0.7 Co 0.3 Ge (001) (at 120 nm and 160 nm) thin plates, respectively.The SkX phase widens the T-H window as the sample thickness decreases, indicating increased stability of the SkX in thinner plates.These results suggest that when the sample thickness is below λ, the magnet can be considered a two-dimensional system, while thickness exceeding 2λ results in a three-dimensional system.The stability of the SkX is reasonably enhanced with decreasing the dimensionality of helimagnets [9,22,29,42].

Summary and conclusions
In summary, our comprehensive cryogenic LTEM study on magnetic textures in Si-and Co-doped FeGe has deepened our understanding of their magnetic characteristics and the stability criteria for the SkX state.The doped FeGe samples exhibit lower T C (approximately 180 K for FeSi 0.3 Ge 0.7 and ∼250 K for Fe 0.7 Co 0.3 Ge) compared to non-doped FeGe (∼280 K).Furthermore, the choice of doping elements impacts the λ in Fe 0.7 Co 0.3 Ge, which is approximately twice as long as that in FeSi 0.3 Ge 0.7 .The formation of the SkX state in Si-and Co-doped FeGe reveals different orientation preferences: the SkX is less stable in the FeSi 0.3 Ge 0.7 (111) plate, while it is more robust in the Fe 0.7 Co 0.3 Ge (111) thin plate.Additionally, our results emphasize that FC prior to LTEM observation can stabilize the metastable SkX state, even in the absence of an external magnetic field.Moreover, we have demonstrated the thickness dependence of the SkX state stability for both FeSi 0.3 Ge 0.7 and Fe 0.7 Co 0.3 Ge thin plates, revealing the robust SkX state in thinner samples.These findings offer new insights into understanding the underlying physics of the topological mechanisms for the skyrmion formation.

Figure 1 .
Figure 1.Overview of magnetic properties in doped FeGe compounds.(a)-(b) The magnetic transition temperature (TC) and wavelength (λ) versus the doping level (x/y) in Fe 1−x CoxGe and FeSiyGe 1−y .Insets depict schematics of crystal structure of Coor Si-doped FeGe.Open squares specify data points in the present work, whereas solid squares, triangles, and circles show data points reproduced from [23, 27, 32, 43].Dashed lines serve as guidelines.

Figure 2 .
Figure 2. Lorentz transmission electron microscopy (LTEM) images observed in FeSi0.3Ge0.7 (a), (d), (g) (001), (b), (e), (h) (110), and (c), (f), (i) (111) thin plates.The defocus value is −240 µm.(a)-(c) LTEM images observed at zero field and 100 K. (d)-(f) LTEM images captured at 70 mT and 170 K.The thickness for both thin plates is approximately 170 nm.(g)-(i) Magnetization maps of one unit cell of SkX squared in (d)-(f), respectively, obtained by transport-of-intensity (TIE) analyses.The scale bars in (a) and (g) represent the scale for LTEM images (a)-(f) and corresponding magnetization maps (g)-(i), respectively.White arrows in (a)-(c) indicate the direction of Q vectors.Insets at the lower left corners of (d)-(f) present the fast Fourier transforms (FFT) of LTEM images.The colors and white arrows in figures.(g)-(i) show the magnitude and direction of in-plane magnetizations scaling a color wheel, as shown on the left side of (g).
f) show the SkX state observed at 70 mT.The corresponding fast Fourier transforms (FFT) of the LTEM images (shown in the lower-left corner inserts) indicate the hexagonal symmetry of the SkX within thin plates.To characterize the skyrmion helicity in FeSi 0.3 Ge 0.7 thin plates, the TIE analysis is employed using underfocused and overfocused LTEM images.The corresponding in-plane magnetization maps in the (001), (110), and (111) oriented thin plates are presented in figures 2(g)-(i), respectively.The Fe 0.7 Co 0.3 Ge sample also shows the evolution of helical structures into SkX upon a normal magnetic field of 70 mT at 220 K, as shown in figure S1.However, the λ in Fe 0.7 Co 0.3 Ge is around 103 nm, which is approximately two times longer than that in FeSi 0.3 Ge 0.7 .Notably, opposite skyrmion helicities have been observed in Fe 0.7 Co 0.3 Ge: counterclockwise helicity for skyrmions in the (111) orientation, and clockwise helicity for skyrmions in the (

Figure 4 .
Figure 4. Phase diagrams of the magnetic structures observed by LTEM in the temperature (T)-magnetic field (H) plane.Zero-field cooling (ZFC) phase diagrams in (a)-(c) and (e)-(g) are for (001), (110), and (111) thin plates of FeSi0.3Ge0.7 and Fe0.7Co0.3Ge,respectively.Field-cooling (FC) phase diagrams in (d) and (h) were obtained in thin plates by 146 mT FC (b) and 50 mT FC (e), respectively.The upper right insets in (a) and (e), and in (d) and (h) are drawings to show the process of obtaining the phase diagrams after ZFC and FC processes, respectively.The PM stands for paramagnetic state, respectively.The thicknesses of thin plates in (a)-(d) and (e)-(h) are around 170 nm and 120 nm, respectively.The color bars in (a) and (e) correspond to the skyrmion (S k ) density in FeSi0.3Ge0.7 and Fe0.7Co0.3Ge.The upper right insets in (b) and (f) show LTEM images of SkX in the (110) thin plates.
)-(c) depict the ZFC magnetic phase diagrams for FeSi 0.3 Ge 0.7 at (001), (110), and (111) planes, respectively.These thin plates have a thickness of approximately 170 nm.It is worth noting that the SkX state in the FeSi 0.3 Ge 0.7 sample exhibits a narrower temperature range in the (111) thin plate compared to those in the other two orientated samples, suggesting that the SkX is relatively unstable in the (111) thin plate.Figures 4(e)-(g) show the ZFC magnetic phase diagrams for Fe 0.7 Co 0.3 Ge at (001), (110), and (