Direct observations of spin fluctuations in spin-hedgehog-anti-hedgehog lattice states in MnSi_1-xGe_x ($x=0.6$ and $0.8$) at zero magnetic field

To grasp an overview of the temperature dependence of the magnetic scattering intensity distributed in the $Q-E$ space, we first performed the TOF neutron inelastic scattering experiment at HRC. In Figs. 2 (a)-2(c), we show the intensity maps of the $x=0.8$ sample at three representative temperatures of $T=50,140,$ and $185$ K. At $50$ K, the magnetic Bragg peak is clearly observed at around $q_{\mathrm{m}}\sim 0.26$ Å^-1. As we increase the temperature to 140 K, which is still lower than the critical temperature of this sample ($T_{\mathrm{c}}=150$ K), the magnetic Bragg peak moves to the lower $Q$-region ($q_{\mathrm{m}}\sim 0.19$ Å^-1). In addition, the diffuse scattering significantly develops in the low-Q region, and the diffuse intensity is spreading along both $Q$- and $E$-directions. At $T=185$ K, the Bragg peak completely disappears, but the strong diffuse scattering still remains.

To quantitatively investigate the widths and peak positions of the Bragg and diffuse scattering, we cut the intensity maps into profiles with respect to $Q$ and $E$; the former and the latter are referred to as constant-$E$ and constant-$Q$ profiles, respectively. Figure 2(d) shows the constant-$E$ profile near the elastic condition in which the intensities in the energy range of $-1<E<1$ meV are integrated, measured at 50 K. We observe a sharp peak corresponding to the magnetic long-range order (LRO). The constant-$Q$ profile of the Bragg peak also shows the resolution-limited sharp profile, as shown in Fig. 2(g). These results indicate that there is no spin fluctuations except for possible collective magnon excitations, which were not clearly observed in the present experiment due to the lack of statistics.

The diffuse scattering at 140 K is clearly seen in the constant-$E$ profile shown in Fig. 2(e). Additional intensity emerges in the low-$Q$ region besides the Bragg peak. By fitting two Gaussian functions to the data, the additional component is described by a broad peak located at around $Q=0.16$ Å^-1, which is lower than the position of the Bragg peak. Figures 2(h) and 2(i) show the constant-$Q$ profiles in the $Q$ regions labelled ”I” and ”II” in Fig. 2(e), respectively; both the Bragg and broad peaks contribute to the former, while the latter comprises only of the broad peak. The profile in the region I is well reproduced by a summation of a resolution-limited Gaussian and a broad Lorentzian functions. By contrast, the profile in the region II is fitted just by a Lorentzian function. The broad peak in the constant-$E$ profile is also observed above $T_{\mathrm{c}}$, and it also has a Lorentzian shape as a function of $E$, as shown in Figs. 2(f) and 2(j), respectively. These data demonstrated that the broad peak emerging in the low-$Q$ region correspond to the diffusive spin fluctuations. Note that the diffuse scattering near $T_{\mathrm{c}}$ was also reported in previous small-angle neutron scattering (SANS) studies of MnGe[16, 29]. The present data show that the $x=0.8$ sample exhibits a similar spin dynamics to that of pure MnGe.

We also performed a neutron inelastic scattering experiment on the $x=0.6$ sample in the same manner as that for the $x=0.8$ sample, as shown in Appendix A. In Figs. 3(a)-3(b), we summarize the temperature dependence of the integrated intensities of the Bragg and diffuse scatterings derived from constant-$E$ profiles near the elastic conditions as well as the instrinsic widths of the diffuse scattering at lower $Q$ regions ($0.10<Q<0.15$ Å^-1), the intrinsic width is approximated as ${W}=\sqrt{W_{\mathrm{obs}}^{2}-W_{\mathrm{res}}^{2}}$, where $W_{\mathrm{obs}}$ and $W_{\mathrm{res}}$ are the observed and resolution-limited widths, respectively. As shown in Figs. 3(a)-(b), we can see that both samples show the similar behavior i.e. the LRO components are realized at low temperatures, the LRO and diffuse components coexist at the middle range of temperatures, and only diffuse components exist at high temperatures. The static (S), partially fluctuating (PF), and fully fluctuating (FF) regions for both samples can be separated. The temperature evolution of the intrinsic widths [Figs. 3(c)-(d)] shows that the diffuse components undergo the critical slowing down at the critical temperature $T_{\mathrm{c}}$.

As the Bragg peaks grow with decreasing temperature, the observed intensities are dominated by the static components of the magnetic moments in the systems, and thus the fractions and characteristic time of the fluctuating spin components become rather difficult to evaluate at low temperatures. Specifically, in the $x=0.8$ ($x=0.6$) sample, the constant-$E$ profile at $T=100$ K ($T=60$ K) is slightly broader than the profiles at low temperatures. Hence, it is difficult to resolve the spin fluctuations from the constant-$Q$ profile. To unambiguously determine the temperature dependence of the fraction of the fluctuating spin components, we thus performed the MIEZE- type neutron spin echo spectroscopy for both the samples.

Figures 4(a)-4(d) are the intensity profiles as a function of momentum $Q$ in the $x=0.8$ sample at four representative temperatures of $T=50,100,130,$ and $150$ K measured at VIN ROSE. These profiles were obtained by extracting the intensities measured by $\lambda=5.2\pm 0.3$ Å from the polychromatic TOF scattering data. Similar to the results at HRC, we observed the well-defined peaks attributed to the long-range orders at low temperature and get broadened at high temperature. To distinguish the contributions from the static and fluctuating components, the integration ranges of $Q$ are selected for each representative temperatures. From the selected integration ranges of $Q$, the intensity profiles of $I(Q,t)$ as a function of the Fourier time $t$ are acquired [Figs. 4(e)- 4(h)]. To extract the static/fluctuating fractions and the characteristic time of the fluctuation, we performed fitting analysis using an exponential decay function and a constant term,

At low temperature of $T=50$ K [Fig. 4(e)], the observed peak is clearly static (S) since the $I(Q,t)$ profile is flat in time domain where the static fraction $S_{0}=~1$ (no fluctuating spins at all). In the middle range of temperatures, we observe the coexistence of the static and fluctuating components or partially fluctuating (PF). At $T=130$ K, we deduce two $I(Q,t)$ functions corresponding to the $Q$-ranges labeled I and II, revealing that the $I(Q,t)$ function in the region I remains finite in the $t\rightarrow\infty$ limit, while the $I(Q,t)$ function in the region II decays to zero. The system becomes fully fluctuating (FF) when the temperature reaches $150$ K [Fig. 4(h)] indicated by the decaying curve without the static fraction. These observations are consistent with the HRC results. Importantly, the $I(Q,t)$ function at 100 K slightly deviates from unity. This result unambiguously shows that the spin fluctuations still remain at this temperature.

We also performed MIEZE-type neutron spin echo measurements on the $x=0.6$ sample, as shown in Appendix B. The static fractions as a function of temperature for both $x=0.8$ and $x=0.6$ samples are summarized in Figs. 5(a)-(b). It is shown clearly that the fluctuating spins start to develop around $T_{\rm{s}}=60$ K and $T_{\rm{s}}=50$ K for the $x=0.8$ and $x=0.6$ samples, respectively. The magnetic long-range order and the fluctuating spins coexist in the middle ranges of temperatures of $T_{\rm{s}}<T<T_{\rm{c}}$, where the critical temperatures ($T_{\rm{c}}$) are obtained as $150$ K and $80$ K for the Ge-concentrations of $x=0.8$ and $x=0.6$, respectively. These results support the experimental data measured at HRC shown in Fig. 3.

We show the temperature dependence of the topological Hall resistivity at $\mu_{0}H=1.5$ T (much lower than the critical field) in order to see the sign reversal behavior [Figs. 5(c)-(d)]. By comparing with Figs. 5(a)-(b), we observed the spin fluctuations in a wide range of temperature centered at $T_{\mathrm{c}}$, which coincides with the temperature range where the positive Hall resistivity was observed. These results confirm the theoretical model that the conduction electrons are scattered by the fluctuating spin clusters with a non-zero average of sign-biased scalar spin chirality [19]. As mentioned in the introduction, the $x=0.6$ sample exhibits positive Hall resistivity not only near the critical temperature but also at low temperatures. The present results clearly show that there is no spin fluctuations at the lowest temperature in the $x=0.6$ sample, indicating that the fluctuating spin cluster model cannot be applied to the low-temperature regime of the $x=0.6$ sample. Note that, in MnSi_1-xGe_x compound, the isovalent substitution of Si for Ge atoms does not change the number of charge carrier thus the electronic Fermi level remains unchanged. The Si-substitution will result in change of the lattice constant due to the chemical pressure and also the magnetic modulation length [21, 30], which may modify the Fermi surface due to the nesting-type instability of the magnetic modulation vector-$q$. A sign change of the topological Hall effect by the Fermi surface modification at low temperatures had been studied rigorously in Mn_1-xFe_xSi system [31]. It is challenging for the theory side to confirm this interesting scenario in MnSi_1-xGe_x, as well. This scenario is beyond our experimental study.

We also found that in both samples, the diffuse scattering still remains even at 300 K as shown in Fig. 2, while the topological Hall resistivity disappears above 200 K. In general, in the paramagnetic phase, the spin correlation length and the characteristic time of the spin fluctuation become shorter with increasing temperature. Therefore, the disappearance of the topological Hall resistivity could be interpreted that the spin fluctuations above 200 K no longer contain the spin cluster with a non-zero average of sign-biased scalar spin chirality. We note here that the temperature dependence of the vector spin chirality, $S_{i}\times S_{j}$, in the paramagnetic phase of MnSi was studied by polarized and unpolarized neutron scattering [32]. Just above the critical temperature, MnSi exhibits diffuse scattering arising from the short-range spin correlation, in which the sign of $S_{i}\times S_{j}$ is completely fixed. As the temperature is increased, the chiral spin correlation rapidly decays, and thus the system exhibits non-chiral spin fluctuations at high temperatures. The temperature evolution agrees with the energy scale of Dzyaloshinskii-Moriya (DM) interaction in MnSi derived from the neutron scattering experiment of about $1$ meV [33, 34]. We suggest that the scalar spin chirality in the paramagnetic phase of MnSi_1-xGe_x also has a similar tendency with respect to temperature. Furthermore, the magnitude of DM interaction in MnGe varies depending on the evaluation, the energy scale is less than $\sim 10$ meV [35, 36]. Thus, it is rather reasonable to suppose that the chiral fluctuations in MnSi_1-xGe_x disappear above $\sim 200$ K.

Next, we also estimated the temperature dependence of the characteristic times of spin fluctuations $\tau$ which can be extracted directly from the MIEZE-NSE measurements [Figs. 6(a)-6(b)], it was found in the order of picosecond (ps). We also present the approximate characteristic times $\tau$ derived from the obtained intrinsic energy widths in the HRC experiment, $\tau=\hbar/\gamma$, where $\gamma$ is the half of the intrinsic width $W$ (see Figs. 3(c)-(d)). Although the intrinsic width $W$ is only quite roughly estimated from HRC experiment, it is in good agreement with the characteristic time $\tau$ deduced from the MIEZE-NSE measurement (VR).

The obtained the characteristic times of spin fluctuations $\tau$ near $T_{\mathrm{c}}$ in the $x=0.8$ and $x=0.6$ samples are 12 and 9 ps, respectively. These characteristic times are sufficiently longer than the time scale of the electrons hopping among the atomic sites, which is typically estimated to be in the order of femtosecond from the Fermi velocity and inter-atomic distances [37]. Therefore, the scalar spin chirality in the locally correlated spin-cluster can persists during the scattering process of the electrons.

The present doped samples have shorter characteristic time compared to the pristine MnSi and MnGe. The characteristic times in MnSi and MnGe near $T_{\mathrm{c}}$ are 1 ns [32, 38, 39, 40, 28] and 20 ps [20], respectively. These results suggest that the characteristic time of the spin fluctuations as a function of $x$ shows a non-monotonous behavior. This characteristic time may be related to the magnetic modulation length since such a non-linear behavior was also observed in the magnetic modulation length as given in Ref. [21]. Please note that, at low concentration $x$, the magnetic modulation length is well described by the ratio of the conventional symmetric exchange to DM-type antisymmetric exchange interactions [30]. However, those interactions failed to explain the magnetic structure at higher concentrations ($x>0.25$) where the SHAH lattice states stabilized [21]. The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is considered to be the origin of the SHAH states [41, 42].