Quantum Griffiths phase in disordered ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$

We show the presence of magnetic rare regions consistent with the quantum Griffiths phase in Fe-doped MnSi using detailed heat capacity, magnetization, and muon spin relaxation (μSR) measurements down to millikelvin temperatures. The slow dynamics of these rare regions at low temperatures leads to the non-Fermi-liquid behavior in heat capacity and magnetization. The μSR and magnetization results further indicate that the dynamics freezes into a cluster-glass state below $T_f\sim 1.25$ K. The results are in agreement with theoretical models proposed in the literature for metallic systems with Heisenberg symmetry that exhibit the quantum Griffiths phase in the presence of strong disorder.

Figure 1

(a) $\chi \left(T\right)$ and (b) $\mathrm{\Delta }C\left(T\right)/T$ on a log-log scale. The vertical dash-dotted and dotted lines denote the transition to freezing temperature $T_f$ and precursor state $T^{\prime }$ corresponding to the maxima and minima in $d\chi \left(T\right)/dT$ [inset of (a)] respectively. (c) $M$ vs $H$ at three temperatures. Inset: $M-H$ hysteresis loop measured in low $H$ at 500 mK. The black lines in (a), (b), and (c) are fit to a power-law divergence. (d) Scaling of $\left(M/H\right)T^{\eta }$ vs $H/T^{\delta }$. Inset: Arrott plots at 0.5, 2.0, and 10 K.

Figure 2

(a)–(c) $\mathrm{ZF}-\mu \mathrm{SR} A\left(t\right)$ at a few representative temperatures from 40 to 75 mK. Inset of (c) shows $A\left(t\right)$ at short times. (d) $\mathrm{LF}-\mu \mathrm{SR} A\left(t\right)$ at $T=1.7$ K for different $H$. The solid lines are the fits to the data using Eq. (3). Inset: ${\lambda }_2\left(H\right)$ obtained using Eq. (3) and the red line is a fit.

Figure 3

(a) $f\left(T\right)$, (b) ${\lambda }_2\left(T\right)$, (c) ${\lambda }_1\left(T\right)$, and (d) $\beta \left(T\right)$, down to 75 mK obtained from fits of $\mathrm{ZF}-\mu \mathrm{SR}$ spectra using Eq. (3). The solid red line in (b) is a fit as described in the text. The transition to $T_f$ and $T^{\prime }$ is denoted by vertical dash-dotted and dotted black lines in accordance with the $\chi \left(T\right)$ and $\mathrm{\Delta }C\left(T\right)/T$, respectively.

Figure 4

Schematic illustration of $T-x$ or $p-x$ phase diagram with increasing disorder strength from (a) to (c) according to Ref. [8]. (a) Clean systems: resulting in FO QPT above TCP. (b) Intermediate disorder: resulting in a QCP. (c) Strong disorder: showing the effect of rare regions for Heisenberg symmetry [11, 12, 13]. Inset: (d) representing the survival of clusters and their freezing. (e) $T-x$ phase diagram for ${\mathrm{Mn}}_{1-x}{\mathrm{Fe}}_x\mathrm{Si}$ showing the vanishing of $T_f$ at $x^{**}$. Few data points are taken from Refs. [22, 25, 27].