Ferroquadrupolar ordering in a magnetically ordered state in ErNiAl

We conducted ultrasonic measurements to clarify whether a phase transition exists in ErNiAl, a hexagonal compound, below the antiferromagnetic transition temperature, ${T}_{\mathrm{N}}\ensuremath{\sim}6$ K. We discovered a significant elastic softening of the transverse modulus, ${C}_{66}$, accompanied by a significant ultrasonic attenuation toward ${T}_{\mathrm{Q}}=3.4$ K, which is the temperature of a sharp downward peak in other moduli, indicating a phase transition at ${T}_{\mathrm{Q}}$. The crystal field analysis reveals that the softening of ${C}_{66}$ below 80 K is due to an interlevel ${O}_{xy}$-type quadrupole interaction with a positive quadrupole-quadrupole coupling constant between the ground and excited Kramers doublets. A spontaneous expectation value of ${O}_{xy}$ emerges at ${T}_{\mathrm{Q}}$ in our crystal field model with the mean-field approximation. No quadrupolar ordering occurs in a magnetically ordered state because the degeneracy of quadrupoles is lifted by an internal magnetic field of magnetic ordering. However, our experimental and calculated results suggest that the driving force of the phase transition at ${T}_{\mathrm{Q}}$ is ${O}_{xy}$-type ferroquadrupolar ordering, implying a quadrupolar ordering in a magnetically ordered state at zero magnetic field.