Non-Fermi-Liquid Behavior within the Ferromagnetic Phase in ${\mathrm{U}\mathrm{R}\mathrm{u}}_{2-x}{\mathrm{R}\mathrm{e}}_x{\mathrm{S}\mathrm{i}}_2$

The ${\mathrm{U}\mathrm{R}\mathrm{u}}_{2-x}{\mathrm{R}\mathrm{e}}_x{\mathrm{S}\mathrm{i}}_2$ system exhibits ferromagnetic order for Re concentrations $0.3<x\le 1.0$. Non-Fermi-liquid (NFL) behavior is observed in the specific heat for $0.15\le x\le 0.6$ [$C/T\propto -\ln T$ (or $T^{-0.1}$)], and also in the power-law $T$ dependence of the electrical resistivity [$\rho (T)\propto T^n$] with $n<2$ for $0.15\le x<0.8$, at low $T$, providing strong evidence that the NFL behavior persists within the ferromagnetic phase. Furthermore, the deviation of the physical properties of ${\mathrm{U}\mathrm{R}\mathrm{u}}_{2-x}{\mathrm{R}\mathrm{e}}_x{\mathrm{S}\mathrm{i}}_2$ from Fermi-liquid behavior is most pronounced at the ferromagnetic quantum critical point, and the NFL behavior found in the ferromagnetic phase may be consistent with the Griffiths-McCoy phase model.

Figure 1

Magnetic phase diagram of ${\mathrm{U}\mathrm{R}\mathrm{u}}_{2-x}{\mathrm{R}\mathrm{e}}_x{\mathrm{S}\mathrm{i}}_2$ showing the antiferromagnetic or hidden order (AFM or HO, filled circles) [19], superconducting (SC, open circles) [19], and ferromagnetic [FM, filled squares [20]: Arrott plot analysis; filled diamonds: modified Arrott plots; open squares: low-field $\chi (T)$ data] regions. The power-law exponent of $\rho (T)$, $n$, is shown in the top part of the diagram.

Figure 2

(a) Modified Arrott plot $M^{1/\beta }$ vs $(H/M{)}^{1/\gamma }$ for $x=0.35$. (b) $M$ vs $H$ of ${\mathrm{U}\mathrm{R}\mathrm{u}}_{2-x}{\mathrm{R}\mathrm{e}}_x{\mathrm{S}\mathrm{i}}_2$ for $x=0.3$. The lines are fits to $M=H^{1/\delta }$. Inset: $\delta (0)$ vs $x$ obtained from a linear extrapolation of the $\delta (T)$ data (closed symbols) or by the modified Arrott plot analysis (open symbols) discussed in the text.

Figure 3

Specific heat $C$ divided by temperature $T$ vs $T$ for $0\le x\le 1.0$ on a semilog scale. The solid (dashed) lines are linear (power-law) fits to the data. Data for the $x=1$ sample are from Ref. 20. Inset: Coefficient of the logarithmic contribution $c_0$ vs $x$. The dashed line is a guide to the eye. The location of FM ($x_c^{\mathrm{F}\mathrm{M}}=0.3$) QCP is indicated by the vertical dotted lines.

Figure 4

$[\rho -{\rho }_0]/\rho (300\mathrm{K})$ vs $T$ for $0.15\le x\le 0.7$, plotted on a log-log scale. The solid lines are power-law fits to the data yielding the exponents $n$. Each of the curves has been shifted up by one decade from the curve below it for clarity. Inset: $\rho /\rho (300\mathrm{K})$ vs $T$ of ${\mathrm{U}\mathrm{R}\mathrm{u}}_{2-x}{\mathrm{R}\mathrm{e}}_x{\mathrm{S}\mathrm{i}}_2$ for $0.2\le x\le 0.8$.