Direct observations of spin fluctuations in hedgehog–anti-hedgehog spin lattice states in MnSi1−xGex (x=0.6 and…

The helimagnetic compounds ${\mathrm{MnSi}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}$ show the three-dimensional multiple-$q$ order referred to as hedgehog--anti-hedgehog spin lattice (HSL) states. Two representative forms of HSL are the cubic-$3q$ spin lattice with $q\ensuremath{\parallel}\ensuremath{\langle}100\ensuremath{\rangle}$ and the tetrahedral-$4q$ spin lattice with $q\ensuremath{\parallel}\ensuremath{\langle}111\ensuremath{\rangle}$, which show up typically for $x=1.0--\phantom{\rule{0.16em}{0ex}}0.8$ and 0.6, respectively. Here, we have investigated the spin fluctuations in the ${\mathrm{MnSi}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}$ polycrystalline samples with $x=0.6$ and 0.8 by using the time-of-flight neutron inelastic scattering and modulated intensity with zero effort type of neutron spin-echo techniques to elucidate the microscopic origin of the unconventional Hall effect in the HSL states. This research is motivated by the observation of a sign change in the unconventional Hall resistivity as a function of temperature [Y. Fujishiro, N. Kanazawa, T. Nakajima, X. Z. Yu, K. Ohishi, Y. Kawamura, K. Kakurai, T. Arima, H. Mitamura, A. Miyake, K. Akiba, M. Tokunaga, A. Matsuo, K. Kindo, T. Koretsune, R. Arita, and Y. Tokura, Nat. Commun. 10, 1059 (2019)]. The present results reveal the correspondences between the temperature ranges where the positive Hall resistivity and spin fluctuations are observed. These results agree well with the theoretical model of the conduction electrons scattered by the fluctuating spin clusters with a nonzero average of sign-biased scalar spin chirality as a mechanism of the positive Hall resistivity [H. Ishizuka and N. Nagaosa, Sci. Adv. 4, eaap9962 (2018)].