Critical phenomenon of the near room temperature skyrmion material FeG - PDF document

1 1 Critical phenomenon of the near room t
1 Critical phenomenon of the near room temperature skyrmion material FeGe, WensenWeiChangjin & YuhengThe 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  4.9 which is close to the theoretical prediction of 3D-Heisenberg model (r P231 , 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 nally causes a long modulation period of a helimagnetic ground state. A bulk FeGe sample exhibits a long-range magnetic order at Curie temperature 278.2K, and displays a complex succession of temperature-driven crossovers in the vicinity of 100 direction below 280K. With decreasing temperature, it changes to the 111 direction at 211In 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 03360004 , 13520003 , and 52670001 ) are obtained, where the self-consistency and reliability are veried by the Widom scaling law and the scaling equations. ese critical High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China. University of Science and Technology of China, Hefei 230026, China. Hefei National Laboratory for Physical Sciences at the Microscale, | 6:22397 | DOI: 10.1038/srep22397 www.nature.com/scientificreports 2 behavior of FeGe indicates a short-range magnetic interaction with a magnetic exchange distance decaying as Jrr()49 . e obtained critical exponents also imply an anisotropic magnetic coupling in FeGe system.Results and DiscussionIt 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  leads to universal scaling laws for the spontaneous magnetization and initial susceptibility 0 . Subsequently, the mathematical denitions of the exponents from magnetization are described as   MTMTT()(),0, (1)SC0  ThMTT()(/),0,(2)C0100   MDHTT,0, (3)C1/ where TTT()/CC is the reduced temperature; Mh/00 and are the critical amplitudes. e parameters (associated with MS (associated with 0 ), and  (associated with ) are the critical exponents. Universally, in the asymptotic critical region (01) these critical exponents should follow the Arrott-Noakes equation of state   HMTTTMM(/)()/(/) (4)CC1/11/ erefore, the critical exponents and can be obtained by tting the MT()S and T()01 curves using the modied Arrott plot of M1/ vs HM(/)1/ . Meanwh

2 ile, can be generated directly by the
ile, can be generated directly by the MH() at the critical temperature according to Eq.(3).Generally, the critical temperature can be roughly determined by the temperature dependence of magnetization [ MT() ]. Figure1(a) shows the MT() curves for FeGe under zero-eld-cooling (ZFC) and eld-cooling (FC) with an applied eld Oe. e MT() curves exhibit an abrupt decline with the increase of temperature, corresponding to the paramagnetic-helimagnetic (PM-HM) transition. A sharp peak is observed at 278.5K. e inset of Fig.1(a) gives dMdT/ , where K is determined from the minimum of the dMdT/ curve. Figure 1.) e temperature dependence of magnetization [ MT() ] for FeGe under Oe [the inset shows the derivative magnetization ( dMdT/ ) the isothermal magnetization MH[()] at 4K (the inset gives the magnied region in the lower eld regime). | 6:22397 | DOI: 10.1038/srep22397 3 Wilhelm et al. has demonstrated that a long-rang magnetic order occurs below 278.2K, however, an inhomogeneous helical state has existed above that temperature due to the strong precursor phenomena. e higher determined here indicates the appearance of precursor phenomena which may be caused by the strong spin uctuation. Figure1(b) shows the isothermal magnetization MH() at 4K, which exhibits a typical magnetic ordering behavior. e inset of Fig.1(b) plot the magnied MH() in lower eld regime, which shows that the saturation eld Oe. No magnetic hysteresis is found on the MH() curve, indicating no coercive force for FeGe.Usually, the critical exponents can be determined by the Arrott plot. For the Landau mean-eld model with 05 and 10 , the Arrott-Noakes equation of state evolves into HMABM/2 , the so called Arrott equation. In order to construct an Arrot plot, the isothermal magnetization curves MH() around are measured as shown in Fig.2(a). e Arrott plot of HM/ 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 one30. Therefore, the Arrott plot of FeGe implies a second-order phase transition, in agreement with the specic heat measurement. According to the Arrott plot, the HM/ generally present a series of parallel straight lines around , where HM/ at just pass through the origin. One can see that all HM/ curves show quasi-straight lines with positive slopes in high eld range. However, all lines show an upward curvature and are not parallel to each other, indicating that the 05 and 10 within the framework of Landau mean-eld model is unsatised. erefore, a modied Arrott plot should be employed.Four kinds of possible exponents belonging to the 3D-Heisenberg model (0365 , 1336) , 3D-Ising model (0325 , 124) , 3D-XY model (0345 , 1316) , and tricritical mean-eld model (025 , 10) are used to

3 construct the modied Arrott plots,
construct the modied Arrott plots, as shown in Fig.3(a–d). All these four constructions exhibit quasi-straight lines in the high eld region. Apparently, the lines in Fig.3(d) are not parallel to each other, indicating that the tricritical mean-eld model is not satised. However, all lines in Fig.3(a–c) are almost parallel to each other. To determine an appropriate model, the modied Arrott plots should be a series of parallel lines in the high eld region with the same slope, where the slope is dened as STdMdHM()/(/)1/1/ . e normalized slope () is dened as NSSTST()/()C , which enables us to identify the most suitable model by comparing the with the ideal value of ‘1’. Plots of for the four dierent models are shown in Fig.4. One can see that the of 3D-Heisenberg model is close to ‘1’ mostly above , while that of 3D-Ising model is the best below . is result indicates that the critical behavior of FeGe may not belong to a single universality class. Figure 2.) e initial magnetization around TC for FeGe; () Arrott plots of HM/ [the MH() curves are measured at interval T1 K, and T05 K when approaching | 6:22397 | DOI: 10.1038/srep22397 4 e precise critical exponents and should be achieved by the iteration method. e linear extrapolation from the high eld region to the intercepts with the axes M1/ and HM(/)1/ yields reliable values of MT(,0)S and T(,0)01 which are plotted as a function of temperature in Fig.5(a). By tting to Eqs.(1) and (2), one obtains a set of and . e obtained and are used to reconstruct a new modied Arrott plot. Consequently, new MT(,0)S and T(,0)01 are generated from the linear extrapolation from the high field region. Therefore, another set of and can be yielded. is procedure is repeated until and do not change. As one can see, the obtained critical exponents by this method are independent on the initial parameters, which conrms these critical exponents are reliable and intrinsic. In this way, it is obtained that 03360004 with T28318005C and 13520003 with T28287008C for FeGe. e critical temperature from the modied Arrott plot is in agreement with that obtained from the derivative MT() curve, indicating strong critical uctuation before the formation of the long-range ordering in FeGe. is critical uctuation is in agreement with the precursor phenomenon reported by Wilhelm et al.. e modulated precursor states and Figure 3.e isotherms of M1/ vs HM(/)1/ with () 3D-Heisenberg model; () 3D-Ising model; () 3D-XY model; and () tricritical mean-eld model. Figure 4.e normalized slopes [ NSSTST()/()C ] as a function of temperature. 5 complexity of the magnetic phase diagram near the magnetic ordering are explained by the change of the character of solitonic inter-core interactions and the onset of specic conned chiral modulationsFigure5(b) shows the isothermal magnetizatio

4 n MH() at the critical temperature &#
n MH() at the critical temperature T283C K, with the inset plotted on a lglg scale. One can see that the MH() at exhibits a straight line on a lglg scale for HHS . We determine that 52970001 in the high eld region HH()S . According to the statistical theory, these critical exponents should fulll the Widom scaling law   1 (5) As a result, 50240005 is calculated according to the Widom scaling law, in agreement with the results from the experimental critical isothermal analysis. e self-consistency of the critical exponents demonstrates that they are reliable and unambiguous.Finally, these critical exponents should obey the scaling equations. Two dierent 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  MHfH(,)(/)(6) where f are regular functions denoted as f for TTC and f for TTC . Dening the renormalized magnetization as mMH(,) , and the renormalized eld as hH() , the scaling equation indicates that forms two universal curves for TTC and TTC respectively38,39. Based on the scaling equation mfh[()] the isothermal magnetization around for FeGe is replotted in Fig.6(a), where all experimental data collapse onto two universal branches. e inset of Fig.6(a) shows he vs hm/ , where all MTH curves should collapse onto two independent universal curves. In addition, the scaling equation of state takes another form       HM k H(7) where kx() is the scaling function. Based on Eq.(7), all experimental curves will collapse onto a single curve. Figure6(b) shows the MH1/ vs H1/() for FeGe, where the experimental data collapse onto a single curve, and locates at the zero point of the horizontal axis. e well-rescaled curves further conrm the reliability of the obtained critical exponents.e obtained critical exponents of FeGe and other related materials, as well as those from dierent theoretical models are summarized in Table1 for comparison. One can see that the critical exponent of FeGe is close to that Figure 5.) e temperature dependence of MS and for FeGe with the tting solid curves; () the isothermal MH() at with the inset plane on lglg scale (the solid line is tted). 6 of 3D-Heisenberg model, while 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 Jr() M. E. Fisher et al. have treated this kind of magnetic ordering as an attra

5 ctive interaction of spins, where a reno
ctive interaction of spins, where a renormalization group theory analysis suggests Jr() decays with distance  Jrr()(8)d() where is the spatial dimensionality and is a positive constant. Moreover, there is Figure 6.) Scaling plots of renormalized magnetization vs renormalized eld around the critical temperatures for FeGe (the inset shows the hm/ ) the rescaling of the the MH() curves by MH1/ vs H1/() . CompositiontechniqueRef.FeGeis work3D-Heisenbergtheory3D-XYtheory3D-IsingtheoryTricritical mean-eldtheoryMean-eldtheoryMnSiHallOSeOTable 1. Comparison of critical exponents of FeGe with dierent theoretical models and related materials (MAPmodied Arrott plot; HallHall eect; ACac susceptibility; SCsingle crystal; polycrystal). 7      () dnnnndnGnnn14288(2)(4)(8)12(720)(4)(8) (9) where    ()d and  ()()G 3 dd 2142 2 is the spin dimensionality. For a three dimensional material 3), we have Jrr()(3) . When 2 , the Heisenberg model ( 0365 , 1386 and 48 ) is valid for the three dimensional isotropic magnet, where Jr() decreases faster than . When 3/2 , the mean-eld model (05 , 10 and 30) is satised, expecting that Jr() decreases slower than r45 . From Eq.(9) 19080007 is generated for FeGe, thus close to the short-range magnetic coupling of 2 . Subsequently, it is found that the magnetic exchange distance decays as Jrr()49 , which indicates that the magnetic coupling in FeGe is close to a short-range interaction. Moreover, we get the correlation length critical exponent 07090008 (where / , TTT()/CC0 ), and d(2)1127 0.008. Theory gives that 0115(9) for 3D-Heisenberg model and 0110(5) for 3D-Ising model. erefore, these critical exponents indicates that the critical behavior in FeGe is close to the 3D-Heisenberg model with short-range magnetic coupling. However, the discrepancy of the critical exponents to 3D-Ising or 3D-XY models indicates an anisotropic magnetic exchange interaction.As can be seen from Table1, the critical exponents of FeSi and CuOSeO, which also exhibit a helimagnetic and skyrmion phase transition with similar crystal symmetry, are close to the universality class of the 3D-Heisenberg model, indicating a isotropic short-range magnetic coupling. However, the critical behavior of MnSi belongs to the tricritical mean eld model,48. In macroscopic view, the magnetic ordering in cubic FeGe is a DM spiral similar to the structure observed in the isostructural compound MnSi. However, in microscopic view, the magnetic coupling types in these two helimagnets are dierent. e critical behavior of FeGe is roughly similar to those of FeSi or CuOSeO, except a mag

6 netic exchange anisotropy. In MnSi the s
netic exchange anisotropy. In MnSi the spiral propagates are along equivalent 111 directions at all temperatures below T295C K. However, it has been revealed that the helical axis (vector direction) in FeGe depends on temperature. It is along the 001 direction below 280K, and changes to the 111 direction in a lower temperature range at 211K with the decrease of temperature at zero magnetic eld. is 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 dierent from the magnetocrystalline anisotropy. e magnetocrystalline anisotropy is correlated to the crystal structure, while magnetic exchange anisotropy originates from the anisotropic magnetic exchange coupling J In summary, the critical behavior of the near room temperature skyrmion material FeGe has been investigated around . e reliable critical exponents ( 03360004 , 13520003 , and 52670001 ) are obtained, which are veried by the Widom scaling law and scaling equations. e magnetic exchange distance is found to decay as Jrr()49 , which is close to that of 3D-Heisenberg model (). e critical behavior indicate that the magnetic interaction in FeGe is of short-range type with an anisotropic magnetic exchange coupling.A polycrystalline B20-type FeGe sample was synthesized with a cubic anvil-type high-pressure apparatus. e detailed preparing method was described elsewhere, and the physical properties were carefully checked [H. Du. et al., Nat. Commun. , 8504 (2015)]. e 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. e magnetization was measured using a Quantum Design Vibrating Sample Magnetometer (SQUID-VSM). e no-overshoot mode was applied to ensure a precise magnetic eld. To minimize the demagnetizating eld, the sample was processed into slender ellipsoid shape and the magnetic eld was applied along the longest axis. In addition, the isothermal magnetization was performed aer the sample was heated well above TC for 10 minutes and then cooled under zero eld to the target temperatures to make sure curves were initially magnetized. e magnetic background was carefully subtracted. e applied magnetic eld Ha has been corrected into the internal eld as HHNMa (where is the measured magnetization and is the demagnetization factor) [A. K. Pramanik et al., Phys. Rev. B , 214426 (2009)]. e corrected was used for the analysis of critical behavior.Referencesler, U. ., Bogdanov, A. N. & Peiderer, C. Spontaneous syrmion ground states in magnetic metals. Nature (London)Muuhlbauer, S. et al. Syrmion lattice in a chiral magnet. ScienceMunzer, W. et al. Syrmion lattice in the doped semiconductor FeSi. Phys. ev.

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