Spin-wave stiffness in the Dzyaloshinskii-Moriya helimagnets ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$

The small-angle neutron scattering is used to measure the spin-wave stiffness in the field-polarized state of the Dzyaloshinskii-Moriya helimagnets ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ with $x=0.03$, 0.06, 0.09, and 0.10. The ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ compounds are helically ordered below $T_c$ and show a helical fluctuation regime above $T_c$ in a wide range up to $T_{\text{DM}}$. The critical temperatures $T_c$ and $T_{\text{DM}}$ decrease with $x$ and tend to 0 at $x=0.11$ and 0.17, respectively. We have found that the spin-wave stiffness $A$ change weakly with temperature for each individual Fe-doped compound. On the other hand, the spin-wave stiffness $A$ decreases with $x$ duplicating the $T_{\text{DM}}$ dependence on $x$, rather than $T_c\left(x\right)$. These findings classify the thermal phase transition in all ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ compounds as an abrupt change in the spin state caused, most probably, by the features of an electronic band structure. Moreover, the criticality in these compounds is not related to the value of the ferromagnetic interaction but demonstrates the remarkable role of the Dzyaloshinskii-Moriya interaction as a factor destabilizing the magnetic order.

Figure 1

The temperature dependencies $\chi$ in themagnetic field $H=10$ mT for the ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ compounds with $x=0$, 0.03, 0,06, and 0.09. The inset shows a temperature derivative of the susceptibility $d\chi /dT$ for sample with $x=0.03$. The definitions of $T_C$ and $T_{\text{DM}}$ are given in the text.

Figure 2

Temperature-concentration $\left(T\text{−}x\right)$ phase diagram of the ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ compounds. The helix wave vector $k$ as a function of $x$. Closed symbols give values for the samples of present study. The open symbols are taken from [11, 12] with the systematic shift $\delta x=-0.02$. The dashed lines are guides for the eye.

Figure 3

Maps of the SANS intensities for the single crystal ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ with $x=0.03$ taken at $T=10$ K at the field above $H_{C2}$ (a) $H=0.7$ T, (b) $H=0.9$ T, (c) $H=1.1$ T, and (d) $H=1.8$ T. The arrows show the direction of the field.

Figure 4

The radially averaged scattering intensity centered at the Bragg peak position $I$ vs scattered angle $\theta -{\theta }_B$ for the single crystal ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ with $x=0.03$ taken at $T=10$ K and at $H=1.1$ T and $H=1.8$ T. The broken lines are the results of the fit using Eq. (3). The inset shows the field dependence of the scattering intensity at the Bragg position $I\left({\theta }_B\right)$ at $H>H_{c2}$.

Figure 5

Maps of the SANS intensities for the single crystal ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ with $x=0.03$ taken at the field 1.8 T and (a) $T=4$ K, (b) $T=10$ K, (c) $T=15$ K, and (d) $T=17$ K. The arrows show a field direction.

Figure 6

The radially averaged scattering intensity $I(\theta -{\theta }_B)$ for the single crystal ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ with $x=0.03$ taken at the field 1.8 T and $T=4$, 10, and 15 K. The solid lines are the results of the fit using Eq. (3). The inset shows temperature dependencies of the cut-off angle ${\theta }_C$ and the width $\delta$ obtained from the fit to data at $H=1.8$ T.

Figure 7

Maps of the SANS intensities for the single crystal ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ with $x=0.108$ taken at the field 1.8 T and (a) $T=3$ K, (b) $T=5$ K, and (c) $T=7$ K. The arrows show a field direction.

Figure 8

The radially averaged scattering intensity $I(\theta -{\theta }_B)$ for the single crystal ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ with $x=0.108$ taken at the field 1.8 T and $T=3$, 5, and 7 K. The solid lines are the results of the fit using Eq. (3).

Figure 9

Temperature dependence of the spin-wave stiffness $A$. Dashed lines are fits to a power law (see text).

Figure 10

The spin-wave stiffness $A$ of the ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ compounds versus the characteristic temperatures $T_c$ and $T_{\text{DM}}$.