History dependence of the magnetic properties of single-crystal ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$

We report the magnetization, ac susceptibility, and specific heat of optically float-zoned single crystals of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$, $0.20\le x\le 0.50$. We determine the magnetic phase diagrams for all major crystallographic directions and cooling histories. After zero-field cooling, the phase diagrams resemble that of the archetypal stoichiometric cubic chiral magnet MnSi. Besides the helical and conical state, we observe a pocket of skyrmion lattice phase just below the helimagnetic ordering temperature. At the phase boundaries between these states evidence for slow dynamics is observed. When the sample is cooled in small magnetic fields, the phase pocket of skyrmion lattice may persist metastably down to the lowest temperatures. Taken together with the large variation in the transition temperatures, transition fields, and helix wavelength as a function of the composition, this hysteresis identifies ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ as an ideal material for future experiments exploring, for instance, the topological unwinding of the skyrmion lattice.

Figure 1

Helimagnetism in ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$. (a) Compositional phase diagram. We distinguish a helimagnetic (HM) and a paramagnetic (PM) regime. The transition temperatures determined in this study are consistent with earlier reports. Inset: Low-temperature transition fields as a function of the composition. (b) Helix wavelength, ${\lambda }_{\mathrm{h}}$, and corresponding wave vector, $q_{\mathrm{h}}$, as a function of the cobalt content $x$. Values in (a) and (b) are summarized from reports by Beille et al. [27, 66], Ishimoto et al. [53], Chernikov et al. [34], Chattopadhyay et al. [47], Manyala et al. [29], Onose et al. [39], Grigoriev et al. [49, 50, 55], Takeda et al. [56], and Münzer et al. [3].

Figure 2

Magnetic susceptibility of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$. (a) Temperature dependence of the inverse ac susceptibility. At high temperatures we observe Curie-Weiss-like behavior. Inset: Extrapolated fluctuating moments. For comparison, we show data from Ishimoto et al. [44]. (b) Temperature dependence of the real part of the ac susceptibility, $\mathrm{Re}{\chi }_{\mathrm{ac}}$. Temperatures $T_c$ and $T_2$ are defined as the kink and the point of inflection, respectively.

Figure 3

Field dependence of the magnetization of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ at high fields and low temperatures. (a) Isothermal magnetization at 4 K as a function of the field up to 9 T for different cobalt contents $x$. (b) Arrott plot, i.e., the magnetic field divided by the magnetization versus the square of the magnetization, for $x=0.20$ and several temperatures. Solid lines represent linear fits to the high-field data. Inset: The corresponding isothermal magnetization.

Figure 4

Key properties of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ at high fields and low temperatures. (a) Temperature dependence of the spontaneous moment, $m_s$, as extrapolated from Arrott plots. Dashed lines are fits following $m_s^2=m_{s,0}^2\left(1-{(T/T_{\mathrm{Arr}})}^{\alpha }\right)$. Inset: Concentration dependence of the zero-temperature spontaneous moment, $m_{s,0}$. For comparison, we show data from Beille et al. [27] and Ishimoto et al. [44]. The exponent $\alpha =3/2$ differs from typical itinerant ferromagnets, where one expects $\alpha =2$. (b) Square of the spontaneous moment, $m_s^2$, as a function of $T^{3/2}$.

Figure 5

Typical field and temperature dependence of the susceptibility of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ for various concentrations after zero-field cooling. The magnetic field was parallel $\langle 100\rangle$. (a–d) Field dependence at 2 K (blue curve) and at a temperature a few kelvins below $T_c$ (red curve). The derivative of the dc magnetization, $\mathrm{d}M/\mathrm{d}H$, is shown as gray curve. (e–h) Temperature dependence for zero field (blue curve) as well as for typical field values in the skyrmion lattice state (red curve), the conical state (green curve), and the field-polarized regime (gray curve). See text for definitions of the transition fields and temperatures.

Figure 6

Typical field dependence of the susceptibility of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ with $x=0.20$ for different field directions. Data are shown after zero-field cooling (zfc; open symbols) and high-field cooling (hfc; filled symbols) for 2 K (blue curve) and two temperature just below $T_c$ (green and red curves). (a) Real part of the ac susceptibility, $\mathrm{Re}{\chi }_{\mathrm{ac}}$, for a field along $\langle 100\rangle$. The derivative of the dc magnetization, $\mathrm{d}M/\mathrm{d}H$, is shown as gray curve. Data are offset by 0.25 for clarity. (b) Imaginary part of the ac susceptibility, $\mathrm{Im}{\chi }_{\mathrm{ac}}$, for a field along $\langle 100\rangle$. (c–f) Susceptibility for fields along $\langle 110\rangle$ and $\langle 111\rangle$. The overall behavior is very similar for all major crystallographic directions.

Figure 7

Typical temperature dependence of the susceptibility of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ with $x=0.20$ for different field directions. Data are shown for three applied field values and increasing temperatures after zero-field cooling (zfc; blue curve), field cooling (fc; green curve), and high-field cooling (hfc; red curve) as well as for decreasing temperatures during field cooling-down (fcd; orange curve). (a) Real part of the ac susceptibility, $\mathrm{Re}{\chi }_{\mathrm{ac}}$, for a field along $\langle 100\rangle$. Data are offset by 0.15 for clarity. (b) Imaginary part of the ac susceptibility, $\mathrm{Im}{\chi }_{\mathrm{ac}}$, for a field along $\langle 100\rangle$. Data are offset by 0.005 for clarity. (c–f) Susceptibility for fields along $\langle 110\rangle$ and $\langle 111\rangle$. The overall behavior is very similar for all major crystallographic directions.

Figure 8

Magnetic phase diagram of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ with $x=0.20$ for fields along $\langle 100\rangle$, $\langle 110\rangle$, and $\langle 111\rangle$ after zero-field cooling (left) and field cooling (right). Data are shown as a function of the internal field, i.e., after correction for demagnetization effects. We distinguish six regimes: helical, conical, skyrmion lattice, paramagnetic (PM), field polarized (FP), and fluctuation disordered (FD). The overall behavior is very similar for the three directions. The helical state is only observed after zero-field cooling. Under field cooling the skyrmion lattice may be extended metastably down to low temperatures.

Figure 9

Magnetic phase diagram of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ for various concentrations after zero-field cooling (left) and field cooling (right). The magnetic field was parallel $\langle 100\rangle$. Data are shown as a function of the internal field, i.e., after correction for demagnetization effects. Depending on the cobalt content $x$, the critical temperature and field values vary, while the magnetic phase diagram stays qualitatively the same in the concentration range studied.

Figure 10

Temperature dependence of the specific heat of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$. The magnetic field was applied along $\langle 110\rangle$ after zero-field cooling. (a) Specific heat for different cobalt contents, $x$, and magnetic fields. Data are offset by 2 J ${\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$ for clarity. A Vollhardt invariance at $T_2$ is observed in small fields at all concentrations studied. (b) Specific heat divided by temperature. Data are offset by 40 mJ ${\mathrm{mol}}^{-1}{\mathrm{K}}^{-2}$ for clarity. Inset: The crossing point at $T_2$ in small fields for $x=0.20$.

Figure 11

Contributions to the specific heat and the entropy of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ with $x=0.20$ as a function of the temperature. (a) Contributions to the specific heat divided by the temperature. $C_{\mathrm{meas}}$ is the measured specific heat. $C_{\mathrm{ph}}\propto T^3$ is the phonon contribution according to the Debye model. $C_{\mathrm{el}}=C_{\mathrm{meas}}-C_{\mathrm{ph}}$ is the total electronic contribution. $C_{\mathrm{el}}^{\mathrm{lin}}\propto T$ is the linear electronic contribution expected from Fermi-liquid theory. $\mathrm{\Delta }{C}_{\mathrm{el}}=C_{\mathrm{el}}-C_{\mathrm{el}}^{\mathrm{lin}}$ represents additional magnetic contributions. Inset: Zero-temperature extrapolation of $C/T$, denoted ${\gamma }_0$, as a function of the cobalt content. In addition, we show data from Lacerda et al. [38]. (b) Contributions to the entropy as calculated from (a).