Magnetic ground state and spin fluctuations in MnGe chiral magnet as studied by muon spin rotation

We have studied by muon spin resonance $\left(\mu \mathrm{SR}\right)$ the helical ground state and fluctuating chiral phase recently observed in the MnGe chiral magnet. At low temperature, the muon polarization shows double-period oscillations at short-time scales. Their analysis, akin to that recently developed for MnSi [A. Amato et al., Phys. Rev. B 89, 184425 (2014)], provides an estimation of the field distribution induced by the Mn helical order at the muon site. The refined muon position agrees nicely with ab initio calculations. With increasing temperature, an inhomogeneous fluctuating chiral phase sets in, characterized by two well-separated frequency ranges which coexist in the sample. Rapid and slow fluctuations, respectively, associated with short-range and long-range ordered helices, coexist in a large temperature range below $T_{\mathrm{N}}=170$ K. We discuss the results with respect to MnSi, taking the short helical period, metastable quenched state, and peculiar band structure of MnGe into account.

Figure 1

Magnetic structure of MnGe deduced from neutron powder diffraction as seen along (a) [1,1,$\overline{1}]$ and (b) [0,0,1] crystallographic directions. The latter picture consists of six consecutive unit cells, stacked along the $c$ axis.

Figure 2

Time dependence of the asymmetry measured at short times at $T=10$ K. The solid curve is a fit of Eq. (2) to the data; see text.

Figure 3

ZF-$\mu \mathrm{SR}$ results for MnGe at 10 K. (a) Fast Fourier transforms (FFTs) of the raw data (red area) and of the fitted curve (black symbols) (see Fig. 2). (b) FFTs of the fitted curve now compared with the FFTs of the simulated signal (blue area) and the calculated local-field distribution $D\left(B_{\text{loc}}\right)$ (blue symbols). (c) Calculated local-field distribution $D\left(B_{\text{loc}}\right)$ (blue symbols). Black line is a fit of a shifted Overhauser function [Eq. (3)] to the data. See text for more details.

Figure 4

Determination of the couple of optimized parameters $(x_{\mu },A_{\text{cont}})$: the muon fractional coordinate $x_{\mu }=0.543$ and and the contact coupling constant $A_{\text{cont}}=-0.591$ T ${\mu }_{\mathrm{B}}^{-1}$ optimize the calculated distribution of the internal field as compared with the one measured at 10 K. See text.

Figure 5

Result of the ab initio calculation. (a) Interstitial position candidates for a muon site. (b) Final position resulting from the structural relaxation, showing the three Mn neighbors of each muon site.

Figure 6

(a) Asymmetry measured in zero field and at short times for several temperatures. (b) Average internal field $B_{\mathrm{av}}$ compared with the ordered Mn moment measured by neutron diffraction.

Figure 7

$\mu \mathrm{SR}$ spectra recorded in a 20 G longitudinal field and at selected temperatures (a) above and (b) below $T_{\mathrm{N}}$. Solid curves are fits with Eq. (9), as described in the text.

Figure 8

(a) Fluctuating phase fraction derived from a fit of Eq. (9) to the $\mu \mathrm{SR}$ spectra, in comparison with the paramagnetic fraction deduced from Mössbauer spectroscopy on a ${}^{57}\mathrm{Fe}$-doped MnGe sample [19]. (b) Fast and slow relaxation rates obtained through the same analysis. Inset: Slow fluctuations are observed at a small but finite rate ${\lambda }_{\mathrm{s}}$, down to the lowest temperatures. In all panels, lines are guides to the eye.

Figure 9

(a) Effective initial asymmetry of the $\mu \mathrm{SR}$ signal as a function of temperature. The sharp increase of $A_0$ upon heating reflects the melting of the long-range helimagnetic order. (b) Characteristic relaxation rate ${\lambda }^{*}$ obtained from a fit of Eq. (10) to the data (main panel), along with the stretching exponent $\beta$ (inset). (a),(b) Lines are guide to the eye. (c) Calculated relaxation rates distribution computed with the help of experimentally found values for ${\lambda }^{*}$ and $\beta$ (see text).

Figure 10

Relative frequency shift $({\nu }_0-\nu )/{\nu }_0$ (red squares, left scale) deduced from TF-$\mu \mathrm{SR}$ spectra measured in the paramagnetic regime, in comparison with the macroscopic susceptibility (blue circles, right scale). Both sets of data were measured at 0.4 T. A typical TF-$\mu \mathrm{SR}$ spectrum taken at 240 K is shown in the inset.

Figure 11

(a) Thermal evolution of the fast spin fluctuation frequency ${\nu }_{\mathrm{f}}$, derived from our LF-$\mu \mathrm{SR}$ measurements by virtue of Eq. (14). (b) Slow fluctuations frequency as a function of temperature obtained in the low-temperature limit. Solid line are guides to the eye.