Distinct field-induced ferroquadrupolar states for two different magnetic-field directions in DyNiAl

The hexagonal Dy-based compound DyNiAl undergoes ferromagnetic and antiferromagnetic-type magnetic phase transitions at ${T}_{\mathrm{C}}=30$ K and ${T}_{1}=15$ K, respectively. To investigate the $4f$-electronic state and quadrupole interactions in DyNiAl, we carried out ultrasonic measurements versus temperature and applied magnetic field. The transverse elastic moduli ${C}_{44}$ and ${C}_{66}$ show a prominent elastic softening originating from an interlevel ferroquadrupolar-type interaction between the ground state and excited Kramers doublets, clarified by a crystal field analysis. In magnetic fields applied along the [100] and [001] axes, we observed a field-induced phase transition. Because the quadrupole interaction is enhanced in high magnetic fields according to our calculations, we suggest a magnetic-field-induced ferroquadrupolar ordering of the electric quadrupoles ${O}_{xy}$ and ${O}_{yz}$ for fields applied along [100] and [001], respectively, with different quadrupolar order parameters depending on the field direction.