Evolution of Critical Scaling Behavior near a Ferromagnetic Quantum Phase Transition

Magnetic critical scaling in ${\mathrm{URu}}_{2-x}{\mathrm{Re}}_x{\mathrm{Si}}_2$ single crystals continuously evolves as the ferromagnetic critical temperature is tuned towards zero via chemical substitution. As the quantum phase transition is approached, the critical exponents $\gamma$ and ($\delta -1$) decrease to zero in tandem with the critical temperature and ordered moment, while the exponent $\beta$ remains constant. This novel trend distinguishes ${\mathrm{URu}}_{2-x}{\mathrm{Re}}_x{\mathrm{Si}}_2$ from stoichiometric quantum critical ferromagnets and appears to reflect an underlying competition between Kondo and ferromagnetic interactions.

Figure 1

Magnetization isotherms for several ${\mathrm{URu}}_{2-x}{\mathrm{Re}}_x{\mathrm{Si}}_2$ samples exhibiting ferromagnetism. As Re concentration increases, the $M(H)$ curves reflect an increased $T_C$ and moment, which is shown on the right in a direct comparison of low-temperature data across the entire range of $x$ investigated. Data shown for $x=0.3$ and $x=0.6$ were measured at 1.8 K and the other data were taken at 2.0 K.

Figure 2

Arrott-Noakes scaling law plots and modified Arrott plots, vertically paired, for several ferromagnetic samples of ${\mathrm{URu}}_{2-x}{\mathrm{Re}}_x{\mathrm{Si}}_2$. In the scaling plots, $|M|/|t{|}^{\beta }$ vs $H/|t{|}^{\delta \beta }$, the $M(H)$ data collapse onto two curves for $T>T_C$ and $T<T_C$. In the modified Arrott plots, $|M{|}^{1/\beta }$ vs $|H/M{|}^{1/\gamma }$, the $M(H)$ data are linearized and evenly spaced in temperature. Each pair of plots yields a consistent set of values of critical exponents and $T_C$.

Figure 3

Concentration dependence of magnetic critical exponents and phase diagram of ${\mathrm{URu}}_{2-x}{\mathrm{Re}}_x{\mathrm{Si}}_2$. Error bars denote the range of values that satisfy the scaling analysis. The exponents $\delta -1$ and $\gamma$, $T_C$, and $M_0$ all extrapolate to zero near $x=0.15$. The $x$ dependence of the exponents $\delta$ and $\gamma$ is well-described by a linear fit, which also holds true for $T_C$ and $M_0$ for $x\le 0.5$; the FM phase boundary drawn is a guide to the eye. The NFL regime, identified in measurements on polycrystals [13, 14], extends into the FM phase. The hidden order phase boundary follows Ref. 13.