Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl

We performed nuclear-magnetic-resonance measurements on itinerant-electron metamagnet UCoAl to investigate the critical behavior of the magnetism near a metamagnetic (MM) critical endpoint (CEP). We derived $c$-axis magnetization $M_c$ and its fluctuation $S_c$ from the measurements of Knight shift and nuclear spin-lattice relaxation rate $1/T_1$ as a function of the $c$-axis external field ($H_c$) and temperature ($T$). We developed contour plots of $M_c$ and $S_c$ on the $H_c$-$T$ phase diagram, and observed the strong divergence of $S_c$ at the CEP. The critical exponents of $M_c$ and $S_c$ near the CEP are estimated and found to be close to the universal properties of a three-dimensional Ising model. We indicate that the critical phenomena at the itinerant-electron MM CEP in UCoAl have a common feature as a gas–liquid transition.

Figure 1

(a) Hexagonal crystal structure of UCoAl composed by the U-Co(1) layer and Co(2)-Al layer alternatively stacking along the $c$ axis. (b) U-Co(1) layer and Co(2)-Al layer from the view of the $c$ axis. When the external field is applied along the $a$ axis, there exist two inequivalent Al sites, marked by a red circle for ${}^{27}$Al($\phi =0^{\circ }$) and a light green circle for ${}^{27}$Al($\phi =\pm$120${}^{\circ }$), where $\phi$ is the angle between the direction of the external field and the EFG second principal axis in the basal plane.

Figure 2

Schematic figures of (a) a MM transition and (b) a gas–liquid transition. Both figures have a CEP at a finite temperature. Below the CEP two phases are separated by the first-order transition line.

Figure 3

Field-swept NMR spectra at $T=4.2$ K in the field applied along the $a$ axis. The ${}^{59}$Co-NMR spectra are shown with dark and light blue for ${}^{59}$Co(1) and ${}^{59}$Co(2) sites, and ${}^{27}$Al-NMR spectra, which split into two sites in the $H$ parallel to the $a$ axis, are shown with red and green for ${}^{27}$Al($\phi =0^{\circ }$) and ${}^{27}$Al($\phi =\pm$120${}^{\circ }$), respectively. Here, $\phi$ is the angle between the direction of the external field and the EFG second principal axis in a basal plane.

Figure 4

(Upper panel) Temperature dependence of Knight shift $K$ in the field ${\mu }_{0}H\sim$ 4.4 T applied along the $a$ axis (red circle) and the $c$ axis (blue circle). (Lower panel) Temperature dependence of magnetization along the $c$-axis $M_c$ evaluated with the above ${}^{27}$Al-NMR Knight-shift results (blue circle) and bulk magnetization along the $c$ axis ($M_c^{\mathrm{bulk}}$) in ${\mu }_0{H}_c=4.0$ T (green open circle) (Ref. 8). $M_c$ and $M_c^{\mathrm{bulk}}$ were well scaled with the hyperfine-coupling constant $A_{\mathrm{hf}}=0.86$ T/${\mu }_{\mathrm{B}}$.

Figure 5

(Upper panel) Temperature dependence of ${\left(T_{1}T\right)}^{-1}$ in the field ${\mu }_{0}H\sim$ 4.4 T applied along the $a$ axis (red circle) and $c$ axis (blue circle). ${\left(T_{1}T\right)}^{-1}$ in the different fields along the $a$ axis are also plotted. (Lower panel) Temperature dependence of magnetic fluctuation $S_c$ along the $c$ axis evaluated with above ${\left(T_{1}T\right)}^{-1}$ results, together with bulk magnetic susceptibility ${\chi }_c^{\mathrm{bulk}}$ in ${\mu }_{0}H\sim 0.1$ T along the $c$ axis (green open symbol) (Ref. 9). $S_c$ and ${\chi }_c^{\mathrm{bulk}}$ are well scaled with each other.

Figure 6

Temperature scanned ${}^{27}$Al-NMR spectra at the angle $\theta ={78}^{\circ }$ (${\mu }_0{H}_c$ $\sim$ 0.9 T) in the first-order transition region. ${}^{27}$Al-NMR spectra colored by blue and red represent PM and FM components, respectively. Between 11 and 7 K both of the PM and FM signals appear, where the PM and FM components coexist.

Figure 7

Temperature scanned ${}^{27}$Al-NMR spectra at the angle $\theta ={68}^{\circ }$ (${\mu }_0{H}_c$ $\sim$ 1.6 T) in the crossover region. ${}^{27}$Al-NMR spectra colored purple continuously shift.

Figure 8

$H_c$ dependence of $M_c$ and $S_c$ at several fixed temperatures. At 4.2 and 10 K, PM and FM components separated by the first-order transition are denoted as square and triangle symbols, respectively. At the other temperatures, data points are denoted as circle symbols. The measurement scans are shown in the schematic $H_c$-$T$ phase diagram by colored arrows. Each color of an arrow corresponds to that of data points.

Figure 9

Temperature dependence of $M_c$ and $S_c$ for several fixed $\theta$ (i.e., $H_c$). At ${\mu }_0{H}_c=0.7$ and 0.9 T, the PM and FM components separated by the first-order transition are denoted as square and triangle symbols, respectively. At other $H_c$, data points are denoted as circle symbols. The measurement scans are shown in the schematic $H_c$-$T$ phase diagram by colored arrows. Each color of an arrow corresponds to that of data points.

Figure 10

(a) Magnetization along the $c$ axis $M_c$ in UCoAl is shown by a contour plot in the $H_c$-$T$ phase diagram. In the phase diagram, the red (blue) circle symbol denotes the point where the FM component starts to appear (disappear) by the first-order transition between the PM to FM state. The surrounded area by the red and blue circle symbols (gray colored region) shows the region where the PM and FM components coexist. The light green triangle symbol denotes the point of the temperature $T_{\max }$ where $M_c$ has a broad maximum in the PM region. The yellow square symbol denotes the point of crossover determined from the maximum of $\partial M_c/\partial T$. The star symbol denotes the CEP determined as (${\mu }_0{H}_c$, $T$)${}_{\mathrm{CEP}}\sim$ (1 T, 12 K). Passing outside the CEP along the purple arrow, $M_c$ changes continuously from the PM to FM states. (b) Magnetic fluctuation along the $c$ axis $S_c$ in UCoAl is shown by contour plot in the $H_c$-$T$ phase diagram.

Figure 11

The power-law fitting of $M_c$ vs $H_c$ on logarithmic scales, where the CEP determined as (${\mu }_0{H}_c$, $T$; $M_c$)${}_{\mathrm{CEP}}$ = (1 T, 12 K; 0.27 T). The fitting provided the critical exponent as $\delta$ $\sim$ 5.4.

Figure 12

The power-law fitting of $M_c$ vs $T$ on logarithmic scales, where the CEP determined as (${\mu }_0{H}_c$, $T$; $M_c$)${}_{\mathrm{CEP}}$ = (1 T, 12 K; 0.27 T). The fitting provided the critical exponent as $\beta$ $\sim$ 0.26. Two cases of $f=29.8$ MHz (${\mu }_{0}H$ $\sim$ 2.7 T) and $f=49.1$ MHz (${\mu }_{0}H$ $\sim$ 4.4 T) provided the almost same fitting result.

Figure 13

The power-law fitting of $S_c$ vs $T$ on logarithmic scales, where the CEP determined as (${\mu }_0{H}_c$, $T$; $M_c$)${}_{\mathrm{CEP}}$ = (1 T, 12 K; 0.27 T). The fitting provided the critical exponent as $\gamma$ $\sim$ 1.2.

Figure 14

Comparison of the critical exponents ($\delta$, $\beta$, $\gamma$) of the present case with those of the known universality classes [mean-field, 2-D Ising, 3-D Ising, 3-D $XY$, 3-D Heisenberg, 2-D marginal quantum critical point (2-D MQCP) and 3-D marginal quantum critical point (3-D MQCP) model].