Quantum Griffiths phase near an antiferromagnetic quantum critical point: Muon spin relaxation study of $\mathrm{Ce}{\left({\mathrm{Cu}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{Ge}}_2$

The antiferromagnetic order in the heavy-fermion compound ${\mathrm{CeCu}}_2{\mathrm{Ge}}_2$ can be suppressed by Co-doping, and at critical composition $x_c=0.6$ ($T_N\to 0$ K) a quantum critical point has been observed. We have performed zero-field (ZF) and longitudinal-field muon spin relaxation ($\mu \mathrm{SR}$) measurements on polycrystalline samples of $\mathrm{Ce}{\left({\mathrm{Cu}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{Ge}}_2$ ($x=0,0.2,0.6,1$) over a temperature range of 100 mK to 10 K and in applied fields from 0 up to 3000 G. Above any ordering temperature, the muon relaxation spectra can be described by a Gaussian-Kubo-Toyabe times exponential line shape. Below the magnetic ordering temperature (i.e., for $x<0.6$), an additional Gaussian relaxation is observed. The zero-field muon relaxation rate suggests the presence of antiferromagnetic ordering below 4 and 0.8 K for $x=0$ and 0.2 samples, respectively. For $x=0.6$, the magnetic order is completely suppressed, and the quantum critical point is accompanied by non-Fermi-liquid behavior, manifested in the power-law divergence of exponential depolarization, i.e., $\lambda \propto T^{0.55}$. The relaxation rate of $x=0.6$ obeys the time-field scaling relation $G_z(t,H)=G_z(t/H^{\gamma })$, which is considered to be a characteristic feature of quantum critical magnetic fluctuations. Furthermore, for $x=0.6$, the exponent of isotherm magnetization, $M\sim H^{\eta }$, and magnetization-field-temperature scaling is consistent with the ZF-$\mu \mathrm{SR}$ data. These results provide strong evidence for the formation of a quantum Griffiths phase near the antiferromagnetic quantum phase transition.

Figure 1

Magnetization as a function of field at $T=2.5$ K for $x=0.6$, showing a linear $\mathrm{behavior},M\sim H$, for $H<20$ kG and a power-law behavior, $M\sim H^{\eta }$, for $H>20$ kG. The inset shows nonuniversal exponent $\eta$ vs $x$.

Figure 2

The magnetization $(M/H)T^{0.56}$ vs $\log (H/T^{1.4})$ for $x=0.6$ and 0.8 (inset). $M$ is expressed in emu/mole, $H$ in G, and $T$ in K.

Figure 3

ZF-$\mu \mathrm{SR}$ time spectra of $\mathrm{Ce}{\left({\mathrm{Cu}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{Ge}}_2$ for $x=0$ and 0.2 measured at different temperatures. The solid lines represent fits to the relaxation functions given by Eq. (2).

Figure 4

Temperature dependence of the effective initial asymmetry ($A=A_1+A_2$), depolarization rate $\lambda$ and $\sigma$ for $x=0$ and 0.2, obtained from the fitting of the $\mu \mathrm{SR}$ spectra using Eq. (2). The insets of (a) and (d) show the asymmetry of slow exponential component ($A_1$) and fast Gaussian component ($A_2$) for $x=0$ and 0.2, respectively.

Figure 5

LF-$\mu \mathrm{SR}$ time spectra of $\mathrm{Ce}{\left({\mathrm{Cu}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{Ge}}_2$ for $x=0$ and 0.2 above and below ordering temperature at different field up to 3000 G.

Figure 6

ZF-$\mu \mathrm{SR}$ spectra of $\mathrm{Ce}{\left({\mathrm{Cu}}_{0.4}{\mathrm{Co}}_{0.6}\right)}_2{\mathrm{Ge}}_2$ in the temperature range $0.1\mathrm{to}2.5$ K. The solid lines represent fits to the relaxation functions given by Eq. (4).

Figure 7

Temperature dependence of relaxation rate $\lambda$ obtained from the fitting of the $\mu \mathrm{SR}$ spectra using Eq. (4). The solid line represents power-law behavior of $\lambda$. The inset shows the temperature dependence of nuclear depolarization rate ${\sigma }_{\text{KT}}$ in the given temperature range.

Figure 8

LF-$\mu \mathrm{SR}$ spectra of $\mathrm{Ce}{\left({\mathrm{Cu}}_{0.4}{\mathrm{Co}}_{0.6}\right)}_2{\mathrm{Ge}}_2$ at $T=100$ mK in the different field up to 1000 G. The inset shows LF-$\mu \mathrm{SR}$ spectra at $T=4$ K in the applied field up to 100 G.

Figure 9

LF dependence of the relaxation rate $\lambda$. It may be noted that the lower field changes could arise from the decoupling of muons with the nuclear moments. The solid lines indicate the fit results obtained with Eq. (5). The inset shows the log-log plot with the solid lines representing power-law behaviors with different exponents.

Figure 10

Time-field scaling of the $\mu \mathrm{SR}$ relaxation function, $G_z\left(t\right)$ vs ($t/H^{\gamma }$), for $\mathrm{Ce}{\left({\mathrm{Cu}}_{0.4}{\mathrm{Co}}_{0.6}\right)}_2{\mathrm{Ge}}_2$ at 100 mK.

Figure 11

ZF-$\mu \mathrm{SR}$ time spectra of ${\mathrm{CeCo}}_2{\mathrm{Ge}}_2$ measured at different temperatures. The upper inset shows LF-$\mu \mathrm{SR}$ spectra up to 50 G, whereas the lower inset shows the temperature dependence of relaxation rate $\lambda$ obtained from the fitting of the $\mu \mathrm{SR}$ spectra using Eq. (4)