Quantum criticality in $\mathrm{Nd}{\mathrm{Fe}}_2{\mathrm{Ga}}_8$ under magnetic field

We studied the magnetism in ${\mathrm{NdFe}}_2{\mathrm{Ga}}_8$ by magnetic susceptibility, specific heat, and resistivity measurements, which show ${\mathrm{Nd}}^{3+}$ ions forming one-dimensional chains along the $c$ axis. At zero field, two magnetic transitions are found at about 15 and 11 K. For field vertical to the $c$ axis, no significant effects on the magnetic orders are found. On the other hand, the two transitions are merged together at about 4 T for field parallel to the $c$ axis and completely suppressed at about 7 T. The suppression of the magnetic order results in quantum critical behaviors of the conventional three-dimensional spin-density-wave type, such as $\rho \sim T^{1.5}$ and $C/T\sim T^{-0.5}$. However, the temperature dependence of the resistivity is replaced by a $T^{0.5}$ behavior at very low temperatures for field larger than 3 T, which is attributed to the field-induced ferromagnetic moment. Our results suggest that there may be couplings between the antiferromagnetic quantum critical fluctuations and the field-induced ferromagnetic fluctuations.

Figure 1

(a) Crystal structure of ${\mathrm{NdFe}}_2{\mathrm{Ga}}_8$. The black lines represent the unit cell. (b) XRD results of ground ${\mathrm{NdFe}}_2{\mathrm{Ga}}_8$ with $R_p=2.31\%$, $R_{wp}=3.21\%$, and $\chi =1.36$.

Figure 2

(a) and (b) Temperature dependence of the inverse magnetic susceptibility for ${\mathrm{NdFe}}_2{\mathrm{Ga}}_8$ and ${\mathrm{LaFe}}_2{\mathrm{Ga}}_8$, respectively. The dashed lines are results fitted by the Curie-Weiss function as described in the main text. (c) The temperature dependence of specific heats of ${\mathrm{NdFe}}_2{\mathrm{Ga}}_8$ and ${\mathrm{LaFe}}_2{\mathrm{Ga}}_8$ at zero field. (d) The temperature dependence of the magnetic specific heat from Nd moments and the corresponding entropy.

Figure 3

(a) and (b) Temperature dependence of the specific heat $C/T$ of ${\mathrm{NdFe}}_2{\mathrm{Ga}}_8$ for $H//c$ and $H\perp c$, respectively. All the data are from ${\mathrm{NdFe}}_2{\mathrm{Ga}}_8$ from now on. Two magnetic transitions at 0 T are labeled as $T_{N1}$ and $T_{N2}$. (c) and (d) Temperature dependence of magnetic susceptibility $\chi =M/H$ for $H//c$ and $H\perp c$, respectively. Only $T_{N1}$ is visible for $H//c$. The symbols and lines are results of the zero-field-cooling and field-cooling measurements, respectively. (e) and (f) Magnetic hysteresis curves for $H//c$ and $H\perp c$, respectively.

Figure 4

(a) Temperature dependence of resistivity at different fields. (b) The $T^{1.5}$ dependence of the resistivity around 7 T. The solid lines are fitted by the linear function. (c) The $T^2$ dependence of the resistivity above 10 T. The solid lines are linear fitting results. (d) The magnetic field dependence of $A$, which is the coefficient of the $T^2$ term for the resistivity. The solid line is fitted by $A_0{(H-H_{\mathrm{QCP}})}^{\alpha }$. (e) Low-temperature $C/T$ at different fields. The solid line is fitted on the 7-T data by a $T^{-0.5}$ function for $T>1.5\mathrm{K}$. (f) Magnetic field dependence of ${\gamma }_0$.

Figure 5

(a) Temperature dependence of the resistivity at 6.8 T. The solid lines are fitted by the $T^{1.5}$ and $T^{0.5}$ functions. (b) Temperature dependence of $\rho -{\rho }_0$ at different fields in the log-log scale, where ${\rho }_0$ is the extrapolated zero-temperature resistivity. The black and red dashed lines are guides to the $T^2$ and $T^{0.5}$ dependence, respectively. (c) Field dependence of the resistivity difference between the increasing and decreasing field processes at difference temperatures. (d) Temperature dependence of the resistivity difference between the ZFC and FC processes at different fields.

Figure 6

Phase diagram of ${\mathrm{NdFe}}_2{\mathrm{Ga}}_8$ for $H//c$. The color map is $\mathrm{\Delta }{\rho }_H/{\rho }_{\mathrm{inc}}$ in logarithmic scale. $T_{N1}$ and $T_{N2}$ are determined from the resistivity and specific-heat measurements, respectively. Hysteresis behaviors exist in the area below $T_{N2}$ and hence $T_{N1}$ above 4 T, which is labeled as “Hys”, $T^{+}$ and $T^{*}$ represent the characteristic temperatures below which the resistivity follows $T^2$ and $T^{0.5}$ dependence, respectively. High-temperature resistivity around 7 T shows $T^{1.5}$ dependence. The dashed lines are guides to the eye.