Universality of the helimagnetic transition in cubic chiral magnets: Small angle neutron scattering and neutron spin echo spectroscopy studies of FeCoSi

We present a comprehensive small angle neutron scattering and neutron spin echo spectroscopy study of the structural and dynamical aspects of the helimagnetic transition in ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ with $x=0.30$. In contrast to the sharp transition observed in the archetype chiral magnet MnSi, the transition in ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ is gradual, and long-range helimagnetic ordering coexists with short-range correlations over a wide temperature range. The dynamics are more complex than in MnSi and involve long relaxation times with a stretched exponential relaxation which persists even under magnetic field. These results in conjunction with an analysis of the hierarchy of the relevant length scales show that the helimagnetic transition in ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ differs substantially from the transition in MnSi and question the validity of a universal approach to the helimagnetic transition in chiral magnets.

Figure 1

SANS results obtained at D33 by applying the magnetic field along the incident neutron beam ($\vec{B}\parallel {\vec{k}}_i$). Characteristic patterns are shown for $B=$ (a) 0, (b) 22, and (c) 43 mT. (d) The temperature dependence of the total scattered intensity for selected magnetic fields. (e) $S\left(Q\right)$ in arbitrary units, deduced by radially averaging the scattered intensity at a magnetic field of 43 mT, for the temperatures indicated. The solid lines indicate the best fits of Eq. (1) to the data.

Figure 2

SANS results obtained at LARMOR by applying the magnetic field perpendicular to the incident neutron beam ($\vec{B}\perp {\vec{k}}_i$). Characteristic patterns are shown for $B$ = (a) 0, (b) 22, and (c) 43 mT. (d) The temperature dependence of the total scattered intensity for selected magnetic fields. (e) $S\left(Q\right)$ in arbitrary units, deduced by radially averaging the scattered intensity at a magnetic field of 0 mT, for the temperatures indicated. The solid lines indicate the best fits of Eq. (1) to the data.

Figure 3

Comparison of a Gaussian, Lorentzian, and a weighted superposition of a Gaussian and a Lorentzian (solid purple line) to fit $S\left(Q\right)$ at $B=0$ mT and $T=43.5$ K.

Figure 4

The elastic fraction $a_{\mathrm{SANS}}$, the correlation length $\xi$, and the pitch of the helical modulation $\ell$ as a function of temperature for the magnetic fields indicated as derived from the measurements on D33 where the field was applied parallel to the neutron beam ($\vec{B}\parallel {\vec{k}}_i$).

Figure 5

Neutron spin echo spectroscopy results. The intermediate scattering function $I(Q,t)$ measured at (a) 0 and (b) 50 mT. The solid lines indicate the fits with the relation provided in Eq. (2).

Figure 6

Comparison of a fit of the intermediate scattering function $I(Q,t)$ to Eq. (2) exponential ($\beta =1$) and a stretched exponential fit ($0\le \beta \le 1$) for $B=0$ mT and $T=43.5$ K.

Figure 7

Obtained parameters from fitting the intermediate scattering function $I(Q,t)$ with the relation provided in Eq. (2). (a) The elastic fraction $a_{\mathrm{NSE}}$, (b) the relaxation time $t_0$, and (c) the stretching exponent $\beta$ as a function of temperature at $B=0$ and $B=50$ mT.

Figure 8

The chiral fraction as a function of $T-T_C$ for both ${\mathrm{Fe}}_{0.7}{\mathrm{Co}}_{0.3}\mathrm{Si}$ and MnSi at $B=0$ mT. The data from MnSi have been adapted from Refs. [21, 36].

Figure 9

Zero-field ac susceptibility as a function of temperature measured at $f=5$ Hz and reproduced from Ref. [33]. The solid line indicates the fit to the relation of Eq. (4). The inset shows the relative difference of the measured data to the fit with Eq. (4).

Figure 10

Total background-corrected scattered intensity obtained by summing the entire detector as a function of temperature at $B=0$ mT for ${\mathrm{Fe}}_{0.7}{\mathrm{Co}}_{0.3}\mathrm{Si}$ and MnSi as measured on LARMOR. MnSi data adapted from [40] and scaled at the highest temperature indicated.