Critical phenomenon of the near room temperature skyrmion material FeGe

The cubic B20 compound FeGe, which exhibits a near room temperature skyrmion phase, is of great importance not only for fundamental physics such as nonlinear magnetic ordering and solitons but also for future application of skyrmion states in spintronics. In this work, the critical behavior of the cubic FeGe is investigated by means of bulk dc-magnetization. We obtain the critical exponents (β = 0.336 ± 0.004, γ = 1.352 ± 0.003 and β = 5.276 ± 0.001), where the self-consistency and reliability are verified by the Widom scaling law and scaling equations. The magnetic exchange distance is found to decay as r−4.9, which is close to the theoretical prediction of 3D-Heisenberg model (r−5). The critical behavior of FeGe indicates a short-range magnetic interaction. Meanwhile, the critical exponents also imply an anisotropic magnetic coupling in this system.

In recently years, skyrmion state, which is a topologically protected nanoscale vortex-like spin structure, has attracted great interest due to its potential application in spintronic storage function [1][2][3][4][5][6][7][8][9][10][11][12] . It has been demonstrated that the skyrmion phase is thermodynamically stable magnetic vortex state in magnetic crystals 13,14 . In addition, writing and deleting single magnetic skyrmion have been realized in PdFe bilayer on Ir(111) surface 15,16 . These findings pave a significant path to design quantum-effect devices based on the tunable skyrmion dynamics. The room-temperature skyrmion materials hosting stable skrymion phase are paid considerable attention 17 . The cubic FeGe belongs to the space group P2 3 1 , in which the non-centrosymmetric cell results in a weak Dzyaloshinskii-Moriya (DM) interaction. The competition of DM interaction between the much stronger ferromagnetic exchange finally causes a long modulation period of a helimagnetic ground state 1,2,18 . A bulk FeGe sample exhibits a long-range magnetic order at Curie temperature T C = 278.2 K, and displays a complex succession of temperature-driven crossovers in the vicinity of T C 19,20 . The skyrmion phase emerges in a narrow temperature range just below T C in the filed range from 0.15 to 0.4 kOe. The existence of the near room temperature skyrmion phase in FeGe, to our knowledge the highest T C in B20 skyrmion compounds, makes it one of the most promising candidate of the next generation spintronic devices. Recently, more stable skyrmion phase has been realized in FeGe thin film, and it has been claimed that the skyrmions can be tuned by the crystal lattice [21][22][23] . On the other hand, multiple and complex magnetic interactions have also been found in FeGe. An inhomogeneous helimagnetic state has been discovered above T C due to the strong precursor phenomena 19,24 . More interestingly, it has been revealed that the helical axis (q-vector direction) orientates depending on temperature. At zero magnetic field, the helical axis is along the < > 100 direction below 280 K. With decreasing temperature, it changes to the < > 111 direction at 211 K 20 .
In view of the potential application and abundant physics in FeGe, a deep investigation of its magnetic exchange is of great importance not only for fundamental physics such as nonlinear magnetic ordering and solitons but also for creation of a basic for future application of skyrmion states and other chiral modulations in spintronics. In this work, the critical behavior of FeGe has been investigated by means of bulk dc-magnetization. The critical exponents (β = .
± . 5 267 0 001) are obtained, where the self-consistency and reliability are verified by the Widom scaling law and the scaling equations. These critical Scientific RepoRts | 6:22397 | DOI: 10.1038/srep22397 behavior of FeGe indicates a short-range magnetic interaction with a magnetic exchange distance decaying as ≈ − . J r r ( ) 4 9 . The obtained critical exponents also imply an anisotropic magnetic coupling in FeGe system.

Results and Discussion
It is well known that the critical behavior for a second-order phase transition can be investigated through a series of critical exponents. In the vicinity of the critical point, the divergence of correlation length ξ leads to universal scaling laws for the spontaneous magnetization M S and initial susceptibility χ 0 . Subsequently, the mathematical definitions of the exponents from magnetization are described as 25,26 : C 0 C is the reduced temperature; M h / 0 0 and D are the critical amplitudes. The parameters β (associated with M S ), γ (associated with χ 0 ), and δ (associated with T C ) are the critical exponents. Universally, in the asymptotic critical region ε < .  Wilhelm et al. has demonstrated that a long-rang magnetic order occurs below 278.2 K, however, an inhomogeneous helical state has existed above that temperature due to the strong precursor phenomena 19,28 . The higher T C determined here indicates the appearance of precursor phenomena which may be caused by the strong spin fluctuation 24 . Figure 1(b) shows the isothermal magnetization M H ( ) at 4 K, which exhibits a typical magnetic ordering behavior. The inset of Fig. 1

(b) plot the magnified M H
( ) in lower field regime, which shows that the saturation field H S ≈ 3000 Oe. No magnetic hysteresis is found on the M H ( ) curve, indicating no coercive force for FeGe. Usually, the critical exponents can be determined by the Arrott plot. For the Landau mean-field model with β = .
1 0 29 , the Arrott-Noakes equation of state evolves into = + H M A BM / 2 , the so called Arrott equation. In order to construct an Arrot plot, the isothermal magnetization curves M H ( ) around T C are measured as shown in Fig. 2

(a). The Arrott plot of M 2 vs H M
/ for FeGe is depicted in Fig. 2(b). According to the Banerjee's criterion, the slope of line in the Arrott plot indicates the order of the phase transition: negative slope corresponds to first-order transition while positive to second-order one 30 . Therefore, the Arrott plot of FeGe implies a second-order phase transition, in agreement with the specific heat measurement 28  0 5 and γ = .
1 0 within the framework of Landau mean-field model is unsatisfied. Therefore, a modified Arrott plot should be employed.
Four kinds of possible exponents belonging to the 3D-Heisenberg model β = .
( 025, γ = . 1 0) 29,32 are used to construct the modified Arrott plots, as shown in Fig. 3 (a-d). All these four constructions exhibit quasi-straight lines in the high field region [33][34][35] . Apparently, the lines in Fig. 3(d) are not parallel to each other, indicating that the tricritical mean-field model is not satisfied. However, all lines in Fig. 3(a-c) are almost parallel to each other. To determine an appropriate model, the modified Arrott plots should be a series of parallel lines in the high field region with the same slope, where the slope is defined as One can see that the NS of 3D-Heisenberg model is close to '1' mostly above T C , while that of 3D-Ising model is the best below T C . This result indicates that the critical behavior of FeGe may not belong to a single universality class. The precise critical exponents β and γ should be achieved by the iteration method 36 . The linear extrapolation from the high field region to the intercepts with the axes are generated from the linear extrapolation from the high field region. Therefore, another set of β and γ can be yielded. This procedure is repeated until β and γ do not change. As one can see, the obtained critical exponents by this method are independent on the initial parameters, which confirms these critical exponents are reliable and intrinsic. In this way, it is obtained that β = .
As a result, δ = . ± . 5 024 0 005 is calculated according to the Widom scaling law, in agreement with the results from the experimental critical isothermal analysis. The self-consistency of the critical exponents demonstrates that they are reliable and unambiguous.
Finally, these critical exponents should obey the scaling equations. Two different constructions have been used in this work, both of which are based on the scaling equations of state. According to the scaling equations, in the asymptotic critical region, the magnetic equation is written as 25 : where ± f are regular functions denoted as + f for > T T C and − f for < T T C . Defining the renormalized magnetization as ε ε ≡ β ( )], the isothermal magnetization around T C for FeGe is replotted in Fig. 6(a), where all experimental data collapse onto two universal branches. The inset of Fig. 6(a) shows he m 2 vs h m / , where all − − M T H curves should collapse onto two independent universal curves. In addition, the scaling equation of state takes another form 25,38 : where k x ( ) is the scaling function. Based on Eq. (7), all experimental curves will collapse onto a single curve. The obtained critical exponents of FeGe and other related materials, as well as those from different theoretical models are summarized in Table 1 for comparison. One can see that the critical exponent γ of FeGe is close to that of 3D-Heisenberg model, while β approaches to that of 3D-Ising or 3D-XY mode, indicating that the critical behavior of FeGe do not belong to a single universality class. Anyhow, all these three models indicate a short-range magnetic coupling, implying the existence of short-range magnetic interaction in FeGe. As we know, for a homogeneous magnet, the universality class of the magnetic phase transition depends on the exchange distance J r ( ). M. E. Fisher et al. have treated this kind of magnetic ordering as an attractive interaction of spins, where a renormalization group theory analysis suggests J r ( ) decays with distance r as 40,41 : where d is the spatial dimensionality and σ is a positive constant. Moreover, there is 41,42 :   (3 ) . When σ ≥ 2, the Heisenberg model (β = . 0 365, γ = . 1 386 and δ = . 4 8) is valid for the three dimensional isotropic magnet, where J r ( ) decreases faster than r −5 . When σ ≤ 3/2, the mean-field model β = .
± . 1 908 0 007 is generated for FeGe, thus close to the short-range magnetic coupling of σ ∼ 2. Subsequently, it is found that the magnetic exchange distance decays as ≈ − . J r r ( ) 4 9 , which indicates that the magnetic coupling in FeGe is close to a short-range interaction. Moreover, we get the correlation length critical exponent ν = .
As can be seen from Table 1, the critical exponents of Fe 0.8 Co 0.2 Si and Cu 2 OSeO 3 , which also exhibit a helimagnetic and skyrmion phase transition with similar crystal symmetry, are close to the universality class of the 3D-Heisenberg model 45,46 , indicating a isotropic short-range magnetic coupling. However, the critical behavior of MnSi belongs to the tricritical mean field model 47,48 . In macroscopic view, the magnetic ordering in cubic FeGe is a DM spiral similar to the structure observed in the isostructural compound MnSi 49 . However, in microscopic view, the magnetic coupling types in these two helimagnets are different. The critical behavior of FeGe is roughly similar to those of Fe 0.8 Co 0.2 Si or Cu 2 OSeO 3 , except a magnetic exchange anisotropy. In MnSi the spiral propagates are along equivalent < > 111 directions at all temperatures below = . T 29 5 C K. However, it has been revealed that the helical axis (q− vector direction) in FeGe depends on temperature. It is along the < > 001 direction below 280 K, and changes to the < > 111 direction in a lower temperature range at 211 K with the decrease of temperature at zero magnetic field 20 . This unique change of helical axis in FeGe may be correlated with the anisotropy of magnetic exchange in this system, since the magnetic exchange anisotropy also plays an important role in determination of the spin ordering direction. In addition, it should be expounded that the magnetic exchange anisotropy is essentially different from the magnetocrystalline anisotropy. The magnetocrystalline anisotropy is correlated to the crystal structure, while magnetic exchange anisotropy originates from the anisotropic magnetic exchange coupling J.

Conclusion
In summary, the critical behavior of the near room temperature skyrmion material FeGe has been investigated around T C . The reliable critical exponents (β = .
± . 5 267 0 001) are obtained, which are verified by the Widom scaling law and scaling equations. The magnetic exchange distance is found to decay as ≈ − . J r r ( ) 4 9 , which is close to that of 3D-Heisenberg model (r −5 ). The critical behavior indicate that the magnetic interaction in FeGe is of short-range type with an anisotropic magnetic exchange coupling.

Methods
A polycrystalline B20-type FeGe sample was synthesized with a cubic anvil-type high-pressure apparatus. The detailed preparing method was described elsewhere, and the physical properties were carefully checked [H. Du. et al., Nat. Commun. 6, 8504 (2015)]. The chemical compositions were determined by the Energy Dispersive X-ray (EDX) Spectrometry as shown in Fig. S1 and Table S I, which shows the atomic ratio of Fe : Ge ≈ 50.52: 49.48. The magnetization was measured using a Quantum Design Vibrating Sample Magnetometer (SQUID-VSM). The no-overshoot mode was applied to ensure a precise magnetic field. To minimize the demagnetizating field, the sample was processed into slender ellipsoid shape and the magnetic field was applied along the longest axis. In addition, the isothermal magnetization was performed after the sample was heated well above T C for 10 minutes and then cooled under zero field to the target temperatures to make sure curves were initially magnetized. The magnetic background was carefully subtracted. The applied magnetic field H a has been corrected into the internal field as =