Quantum Griffiths Phase in the Weak Itinerant Ferromagnetic Alloy ${\mathrm{Ni}}_{1-x}{\mathbf{V}}_x$

We present magnetization ($M$) data of the $d$-metal alloy ${\mathrm{Ni}}_{1-x}{\mathrm{V}}_x$ at vanadium concentrations close to $x_c\approx 11.4\%$ where the onset of long-range ferromagnetic (FM) order is suppressed to zero temperature. Above $x_c$, the temperature ($T$) and magnetic field ($H$) dependencies of the magnetization are best described by simple nonuniversal power laws. The exponents of $M/H\sim T^{-\gamma }$ and $M\sim H^{\alpha }$ are related by $1-\gamma =\alpha$ for wide temperature ($10<T\le 300\mathrm{K}$) and field ($H\le 5\mathrm{T}$) ranges. $\gamma$ is strongly $x$ dependent, decreasing from 1 at $x\approx x_c$ to $\gamma <0.1$ for $x=15\%$. This behavior is not compatible with either classical or quantum critical behavior in a clean 3D FM. Instead it closely follows the predictions for a quantum Griffiths phase associated with a quantum phase transition in a disordered metal. Deviations at the lowest temperatures hint at a freezing of large clusters and the onset of a cluster glass phase.

Figure 1

Temperature-concentration phase diagram of ${\mathrm{Ni}}_{1-x}{\mathrm{V}}_x$ showing ferromagnetic (FM), paramagnetic (PM), quantum Griffiths (GP), and cluster glass (CG) phases. Circles mark $T_c$ found via the Arrott analysis. The gray line is an extrapolation of $T_c(x)$. Also shown are $T_{\max }$ defined by low-field maxima in $\chi (T)$ and $T_{\mathrm{cross}}$ below which frozen clusters dominate $\chi (T)$, leading to superparamagnetism. Inset: Saturation magnetization ${\mu }_{\mathrm{sat}}$ vs $x$ [determined as $M(T<5\mathrm{K},H>1\mathrm{T})$]. Data from Ref. 17 for $x<11\%$ are included.

Figure 2

ac susceptibility ${\chi }_{\mathrm{ac}}$ of the $x=12.25\%$ sample for several dc magnetic fields $H$ and frequencies $\nu$ of the (in phase) ac-magnetic field ($H_{\mathrm{ac}}\approx 0.1\mathrm{G}$) versus temperature $T$. Inset shows the frequency dependence of the maximum in $\chi (T)$ $T_{\max }$. The line follows $d{T}_{\max }/d(\log \nu )=0.018\mathrm{K}/dec(\nu )$.

Figure 3

(a) Low-field susceptibility ${\chi }_m=M/H-{\chi }_{\mathrm{orb}}$ of ${\mathrm{Ni}}_{1-x}{\mathrm{V}}_x$ versus temperature $T$ and (b) low-temperature magnetization $M_m=M-{\chi }_{\mathrm{orb}}H$ versus magnetic field $H$. Dotted lines indicate power laws for $T>10\mathrm{K}$ and $H>3000\mathrm{G}$ in (a) and (b), respectively. Solid lines (shown for $x>12\%$) represent fits to the model (2).

Figure 4

(a) Exponents $\gamma ,1-\alpha$ obtained from ${\chi }_m\sim T^{-\gamma }$ and $M_m\sim H^{\alpha }$ versus V concentration $x$. The plot also shows ${\gamma }_{loT}$ derived from ${\chi }_m(T)$ below 10 K and ${\gamma }_{\mathrm{dyn}}$ extracted from the fit to (2). The $x$-dependence of $\gamma$ can be fitted to a power law, $(1-\gamma )\sim (x-x_c{)}^{\nu \psi }$ (dotted line), as expected according to 12 (with $\nu \psi \approx 0.42$) or to an exponential, $-\ln \gamma \sim (x-x_c)$ (solid line). (b) $H/T$ scaling plot for $x=12.25\%$ showing data at several $T$ from 2–300 K and $H=500\mathrm{G}$. The line is a fit to the form discussed below (1).