Quantum phase transitions in single-crystal ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$: Crystal growth, magnetization, ac susceptibility, and specific heat

We report the magnetization, ac susceptibility, and specific heat of optically float-zoned single crystals of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ for temperatures down to $\sim 2\text{K}$ and magnetic fields up to 9 T. The suppression of the helimagnetic transition temperature $T_1$ above a critical composition $x_1$, as seen in the magnetization, ac susceptibility, and specific heat, suggests the existence of a quantum phase transition at $x_1$. A Vollhardt invariance at a temperature $T_2>T_1$, which may be attributed to the Dzyaloshinsky-Moriya (DM) spin-orbit interactions, is also suppressed with increasing $x$ and vanishes above a concentration $x_2$, where $x_2>x_1$. When suppressing the effects of the DM interactions in an applied magnetic field, the magnetization for sufficiently large fields shares the signatures expected of an underlying putative ferromagnetic quantum critical point for a critical concentration $x_c$, where $x_1<x_c<x_2$. As a function of normalized concentration $x/x_c$, where $x_c^{\text{Co}}\approx 0.084$ and $x_c^{\text{Fe}}\approx 0.192$, the properties of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ are essentially identical with $x_1/x_c\approx 0.78$ and $x_2/x_c\approx 1.17$. Taken together, our study identifies ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ as model systems in which the influence of DM interactions on ferromagnetic quantum criticality may be studied.

Figure 1

(Color online) Single crystals of MnSi, ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$, and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ grown for our study. For each panel the growth direction was from the right to the left. In most crystals a necking was applied at the beginning of the growth to promote grain selection.

Figure 2

Depiction how the $⟨110⟩$-oriented disc was cut up (a $⟨110⟩$ direction was parallel to the line of sight). This way samples for the various bulk properties measured in our study all had the same dimensions. See text for details of layout.

Figure 3

(Color online) Isothermal magnetization of ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ at 4 K as a function of field up to 9 T for various concentrations.

Figure 4

(Color online) Typical magnetic field dependence of the magnetization of MnSi and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ for $x=0.02$ and 0.04. The magnetic field was parallel $⟨100⟩$. The same qualitative behavior is observed in all compounds studied. For clarity only data in the field range $\pm 1\text{T}$ are shown. Data were recorded up to 9 T at selected temperatures, where an unsaturated magnetization is observed down to the lowest temperatures studied.

Figure 5

(Color online) Typical magnetic field dependence of the magnetization of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ for $x=0.04$, 0.08, 0.12, 0.16, and 0.19. The magnetic field was parallel $⟨100⟩$. The same qualitative behavior is observed in all compounds studied. For clarity only data in the field range $\pm 1\text{T}$ are shown. Data were recorded up to 9 T at selected temperatures. The magnetization at high fields is unsaturated down to the lowest temperatures studied.

Figure 6

(Color online) (a) Temperature dependence of the magnetic moment $m_s$ of MnSi, ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$, and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ as extrapolated from the ferromagnetically polarized regime. For all values of $x$ we find a continuous temperature dependence, characteristic of critical transitions. (b) Square of the magnetic moment $m_s^2$ versus temperature squared $T^2$. The linear dependence observed for all $x$ except pure MnSi is a key characteristic of itinerant ferromagnets.

Figure 7

(Color online) (a) Transition temperature as a function of concentration. A sublinear dependence is observed for both ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$. (b) Empirically the suppression is well accounted by a square-root dependence with the concentration. Note the values for the extrapolated critical concentrations given in the figure.

Figure 8

(Color online) Suppression of $m_s$ inferred from the field dependence of the magnetization in the ferromagnetic regime on a normalized concentration scale.

Figure 9

(Color online) Transition temperature $T_c$ versus ordered magnetic moment $m_s^{3/2}$, both inferred from Arrott plots of the magnetization. The proportionality observed here extents over a remarkably wide range and strongly suggests the existence of an underlying ferromagnetic quantum critical point. The data points labelled “$x=0$ pressure” have been inferred from Refs. 57, 90, 91.

Figure 10

(Color online) (a) Temperature dependence of the parameter $A$, the inverse susceptibility, inferred from the magnetization for various compositions of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$. (b) Temperature dependence of the mode-coupling parameter $b$. (c) Extrapolated zero-temperature value of $A$ as a function of normalized concentration $x/x_c$. (d) Extrapolated zero-temperature value of the mode-coupling parameter $b$ as a function of normalized concentration $x/x_c$. The dashed lines are guides to the eyes.

Figure 11

(Color online) (a) Extrapolated temperature $T_b$ of the change in sign from positive to negative of the mode-coupling parameter $b$ inferred from the measured magnetization. (b) Ordered magnetic moment $m_{s,0,ab}$ calculated from the extrapolated zero-temperature value of the phenomenological parameters $A$ and $b$. As a function of $x$ the ordered moment decreases essentially linearly and extrapolates to zero for $x/x_c\to 1$.

Figure 12

(Color online) $B/M$ as inferred from the Arrott plots in the conical phase. This quantity is equivalent to $c{Q}^2$, i.e., the product of spin-fluctuation stiffness with the square of the helimagnetic wave vector $Q$, see Eq. (18).

Figure 13

(Color online) (a) The fluctuation stiffness $c$ inferred from the linear susceptibility in the conical phase (horizontal line in the Arrott plot). Values are given in SI units (note that values given in Ref. 1 are given in cgs units). (b) Concentration dependence of the parameter $\gamma$ describing the energy spread of the relaxation frequency spectra $\Gamma$, where $\gamma$ was inferred from the expression of the Curie temperature in the spin-fluctuation theory, i.e., Eq. (8). Values are given in SI units (note that values given in Ref. 1 are given in cgs units).

Figure 14

(Color online) Temperature dependence of the ac susceptibility of MnSi, ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$, and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$. At high temperatures a pronounced Curie-Weiss dependence is observed. At low temperatures the characteristics of the various modulated spin structures are observed.

Figure 15

(Color online) (a) ac susceptibility as a function of temperature for selected large applied magnetic fields. The maximum at temperature $T_m$ represents the crossover between the field-polarized regime at low temperatures and high fields as opposed to the essentially paramagnetic regime at high temperatures and low fields. (b) Inverse susceptibility of the data shown in panel (a).

Figure 16

(Color online) (a) Temperature dependence of the ac susceptibility of pure MnSi in zero magnetic field (a), $B=0.05\text{T}$ (b), $B=0.1\text{T}$, (c) and $B=0.2\text{T}$ (d). Note the dip in panel (d), which is related to the A phase.

Figure 17

(Color online) (a) Temperature dependence of the ac susceptibility of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ $\left(x=0.04\right)$ in (a) zero magnetic field, (b) $B=0.05\text{T}$, (c) $B=0.1\text{T}$, and (d) $B=0.2\text{T}$. Note the dip in panel (d), which is related to the A phase.

Figure 18

(Color online) Magnetic field dependence of the susceptibility of pure MnSi for different field directions and selected temperatures. Under zfc conditions, shown in panels (a) through (c), the field $B_{c1}$ decreases in the sequences $⟨100⟩$, $⟨110⟩$, and $⟨111⟩$, consistent with the magnetic anisotropy. Under fc conditions, shown in panels (d) through (f), a similar trend may be observed. In the regime of the A phase the susceptibility displays a minimum of decreasing extent in the sequence $⟨100⟩$, $⟨110⟩$, and $⟨111⟩$, again consistent with the magnetic anisotropy.

Figure 19

(Color online) Magnetic field dependence of the susceptibility of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ for $x=0.08$ for different field directions and selected temperatures. Under zfc conditions the field $B_{c1}$ does not change much between $⟨100⟩$, $⟨110⟩$, and $⟨111⟩$, consistent with a reduced magnetic anisotropy as compared with MnSi. Under fc conditions the same trend may be observed, i.e., less anisotropy. Likewise the extent of the signal in the A phase does not change much.

Figure 20

(Color online) Magnetic field dependence of the ac susceptibility of MnSi, ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$, and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ at selected temperatures. Data have been recorded in field cycles from $-1\text{T}$ to $+1\text{T}$ and back, thus reflecting the properties of the field-cooled state.

Figure 21

(Color online) Magnetic phase diagram of MnSi and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ after zero-field cooling and field cooling. We distinguish six regimes: (i) paramagnetism at high temperatures and low fields, (ii) spontaneous helimagnetism (blue shading), (iii) the conical phase (gray shading), (iv) the A phase (red shading), (v) an intermediate regime (purple shading), and (vi) the field-polarized ferromagnetic regime. The shading is extended down to zero temperature, where the lowest temperature studied was $\sim 1.8\text{K}$. The green arrows mark the location of the extrapolated ferromagnetic transition temperature $T_c$. The dashed line marks the location of $T_m$.

Figure 22

(Color online) Magnetic phase diagram of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ after zero-field cooling and field cooling. We distinguish six regimes as explained in the text and the caption of Fig. 21. The phase diagram for field cooling agrees very well with the phase diagram after zero-field cooling, except for the phase boundary of the spontaneous helimagnetic order (blue shading) in all doped samples $\left(x\ne 0\right)$. The shading is extended down to zero temperature, where the lowest temperature studied was $\sim 1.8\text{K}$. The green arrows mark the location of the extrapolated ferromagnetic transition temperature $T_c$. The dashed line marks the location of $T_m$.

Figure 23

(Color online) (a) Curie-Weiss moment of the paramagnetic state as a function of normalized composition $x/x_c$. (b) Ratio of the effective Curie-Weiss moment to the ordered moment inferred from the measured magnetization.

Figure 24

(Color online) Zero-temperature inverse susceptibility as inferred from the Curie-Weiss dependence of the ac susceptibility versus normalized concentration. This quantity extrapolates to zero at the critical concentration $\left(x/x_c\to 1\right)$. The lower dashed line indicates the same property inferred from the field dependence of the magnetization [Fig. 10c], which also extrapolates to zero for $x/x_c\to 1$. The quantitative difference between ac susceptibility and magnetization based data are unexplained.

Figure 25

(Color online) Transition fields versus normalized concentration of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$. The upper critical field appears to collapse towards the critical concentration. For comparison also included are the calculated values of $B_2^{\text{calc}}$ using Eq. (20) (open symbols), which are in remarkable agreement with the values of $B_2$ measured in the ac susceptibility. We distinguish two lower critical fields as determined in zfc and fc conditions. The weak concentration dependence of $B_{1,\text{zfc}}$, which reflects the magnetic anisotropy, could only be determined up to $x/x_c\approx 0.5$. It suggests that the QPT may not be driven by an unpinning of the helical order.

Figure 26

(Color online) Upper critical field squared, $B_{c2}^2$, versus temperature squared. As expected a linear relationship is observed over a wide range. FS and TS denote data points inferred from field and temperature sweeps, respectively.

Figure 27

(Color online) Crossover temperature $T_m$ between the field-polarized ferromagnetic regime and paramagnetic regime for various compositions as plotted on a temperature scale to the 4/3 and a magnetic field scale 2/3. The quasilinear dependence for various $x$ is consistent with ferromagnetic quantum criticality underlying the properties of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$.

Figure 28

(Color online) (a) Temperature versus composition phase diagram summarizing the characteristic transition temperatures inferred from the ac susceptibility. Also shown are the transition temperatures inferred from the magnetization (cf. Fig. 7). (b) Difference $\Delta T_{\text{IM}}$ between the temperatures $T_1$ and $T_2$ as a function of composition. This represents the zero-field temperature range of the intermediate regime in the magnetic phase diagram.

Figure 29

(Color online) Specific heat of pure MnSi and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ as a function of temperature. Panels (a)–(c) show an overview up to 35 K. Panels (d)–(f) show a closeup view of the specific heat of pure MnSi and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ in the vicinity of the magnetic phase transition, highlighting the Vollhardt invariance at $T_2$, and the spike at $T_1$.

Figure 30

(Color online) Specific heat of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ as a function of temperature. Panels (a) through (e) show an overview up to 35 K. Panels (f) through (j) show a closeup view of the specific heat of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ in the vicinity of the magnetic phase transition highlighting the sharp spike at $T_1$ and a Vollhardt invariance at $T_2$.

Figure 31

(Color online) Typical data illustrating the estimate of the lattice contribution of the specific heat in MnSi, ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$, and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$. Data shown were recorded in ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ for $x=0.04$. The experimental data are shown in black, the calculated contribution in the Debye model is shown in green; the cubic low-temperature approximation of the Debye contribution is shown in red. $C_{\text{DP}}$ denotes the Dulong-Petit limit.

Figure 32

(Color online) Debye temperature as a function of normalized concentration. The Debye temperature is essentially constant.

Figure 33

(Color online) Temperature dependence of the electronic contribution of the specific heat divided by temperature, $C_{\text{el}}/T$, in MnSi, ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$, and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ after subtraction of the lattice contribution calculated in a Debye model. For increasing magnetic field $C_{\text{el}}/T$ is suppressed in the low-temperature limit and increases at high temperatures, characteristic of a shift of entropy to high temperatures.

Figure 34

(Color online) Electronic contribution of the specific heat divided by temperature versus temperature on a logarithmic scale. (a) Zero-field data for MnSi and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$. (b) Zero-field data for MnSi and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$.

Figure 35

(Color online) Extrapolated zero-temperature value of $C_{\text{el}}/T$ denoted as ${\tilde{\gamma }}_0$ and the high-temperature contribution ${\tilde{\gamma }}_{\text{fl}}$ observed well above $T_c$ as a function of normalized concentration $x/x_c$.

Figure 36

(Color online) (a) Electronic specific heat divided by temperature of ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ for $x=0.04$, where red data points show the data after subtraction of the lattice contribution. At high temperatures the data are constant in temperature. Data shown in purple represent the low-temperature part after subtraction of the electronic contribution at high temperature. (b) Entropy as calculated for the three electronic contributions to the specific heat shown in (a).

Figure 37

(Color online) (a) Electronic part of the entropy of MnSi and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ as a function of temperature. (b) Magnetic part of the entropy of MnSi and ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ as a function of temperature, i.e., after subtraction of the electronic contribution observed at high temperatures. (c) Electronic part of the entropy of MnSi and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ as a function of temperature. (d) Magnetic part of the entropy of MnSi and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ as a function of temperature, i.e., after subtraction of the electronic contribution observed at high temperatures.

Figure 38

(Color online) Isothermal magnetic entropy released as a function of normalized concentration for different temperatures.

Figure 39

(Color online) Electronic contribution to the entropy at various magnetic fields of MnSi, ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$, and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ inferred from the electronic contribution to the specific heat.

Figure 40

(Color online) Temperature versus concentration phase diagram of ${\text{Mn}}_{1-x}{\text{Co}}_x\text{Si}$ (to the left) and ${\text{Mn}}_{1-x}{\text{Fe}}_x\text{Si}$ (to the right). The suppression of the extrapolated ferromagnetic transition is encompassed by the suppression of the helimagnetic transition and the onset of the intermediate regime, suggesting the possible existence a quantum phase transition at the border of a ferromagnetic quantum critical point followed by a zero-temperature Vollhardt point.