Electric Quadrupolar Contributions in the Magnetic Phases of ${\mathrm{UNi}}_4\mathrm{B}$

We present acoustic signatures of the electric quadrupolar degrees of freedom in the honeycomb-layer compound ${\mathrm{UNi}}_4\mathrm{B}$. The transverse ultrasonic mode $C_{66}$ shows softening below 30 K both in the paramagnetic phase and antiferromagnetic phases down to $\sim 0.33\mathrm{K}$. Furthermore, we traced magnetic field-temperature phase diagrams up to 30 T and observed a highly anisotropic elastic response within the honeycomb layer. These observations strongly suggest that ${\mathrm{\Gamma }}_6(E_{2\mathrm{g}})$ electric quadrupolar degrees of freedom in localized $5f^2$ ($J=4$) states are playing an important role in the magnetic toroidal dipole order and magnetic-field-induced phases of ${\mathrm{UNi}}_4\mathrm{B}$, and evidence some of the U ions remain in the paramagnetic state even if the system undergoes magnetic toroidal ordering.

Figure 1

Crystal and magnetic structure of ${\mathrm{UNi}}_4\mathrm{B}$, reported for $B=0$ [16, 18]. (a) The pseudo-honeycomb network consists of two-thirds of U ions with $Cmcm$ lattice. Red, blue, and green circles indicate U and Ni or B on the layer at $z=0$, $1/2$, respectively. Dashed lines with open circles show the (pseudo) Kagome layer of Ni atoms at $z\sim 1/4$. (b) Colored backgrounds with pink-rhomboidal and blue-rectangular shapes denote antiferromagnetic unit cells in hexagonal ($P6/mmm$) and orthorhombic ($Cmcm$) symmetry, respectively.

Figure 2

Elastic constants of ${\mathrm{UNi}}_4\mathrm{B}$ as a function of temperature. The insets in each panel show enlarged views of the data below 50 K. Illustrations of the distorted hexagon and/or rectangle indicate the lattice strain, which is induced by the respective ultrasonic mode (See Table SI in the Supplemental Material) [21, 25, 26].

Figure 3

(a) CEF level schemes used for the present analysis (see Table SIII, SIV, and SV in the Supplemental Material [21]). Blue and red arrows indicate selection rules where finite matrix elements exist for magnetic dipole $J_x$ and electric quadrupole $O_{xy}$, respectively. (b) Temperature dependence of the elastic constant $C_{66}$, compared with calculations based on the quadrupolar susceptibility using the CEF level schemes shown in panel (a): Scheme 0 (green), scheme 1 (orange), and scheme 2 (blue) as shown by dashed and solid curves. The dotted black and red curves show the background of the elastic constant $C_{66}^0$ in the PM and AFM phases, respectively. The inset shows an enlarged view of the low-temperature region on a log-$T$ scale.

Figure 4

Magnetic-field dependence of $C_{66}$ for (a) $H\parallel [2\overline{1}\overline{1}0]$ and (d) $H\parallel [01\overline{1}0]$ of ${\mathrm{UNi}}_4\mathrm{B}$ at selected temperatures. $C_{66}$ vs temperature under various magnetic fields for (b) $H\parallel [2\overline{1}\overline{1}0]$ and (e) $H\parallel [01\overline{1}0]$. In each panel, vertical arrows indicate elastic anomalies, which correspond to phase boundaries in the magnetic field-temperature phase diagram of ${\mathrm{UNi}}_4\mathrm{B}$ for (c) $H\parallel [2\overline{1}\overline{1}0]$ and (f) $H\parallel [01\overline{1}0]$. Dotted and dashed lines are guides for the eyes. The relative change of the elastic constant $C_{66}$ is highlighted by the color map from zero (red) to large negative (blue) values.