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.

Figure 1

Temperature dependences of the longitudinal elastic moduli $C_{11}$ (left axis) and $C_{33}$ (right axis) in DyNiAl. The vertical arrows indicate the phase transitions at $T_{\mathrm{C}}$ and $T_1$.

Figure 2

Temperature dependences of the transverse elastic moduli $C_{44}$ (left axis) and $C_{66}$ (right axis) in DyNiAl. The vertical arrows indicate the phase transitions at $T_{\mathrm{C}}$ and $T_1$. The inset shows $C_{44}$ in the vicinity of $T_{\mathrm{C}}$. The red solid and blue dashed lines show fit results and the background stiffness, respectively.

Figure 3

(a) The $4f$-level scheme for DyNiAl obtained from the CEF parameters listed in Table 1, where ${\mathrm{\Gamma }}_5$ denotes the irreducible representation for the point symmetry $C_{2v}$. (b) Temperature dependences of the inverse magnetic susceptibility measured in 0.1 T. (c) Magnetization curves at 4.2 K. The calculated results are shown by (b) solid and (c) dashed lines. The magnetization data were taken from Ref. [15].

Figure 4

(a), (b) Temperature and (c), (d) field dependences of the transverse elastic modulus $C_{66}$ for $\mathit{H}\left|\right|\left[100\right]$. We plotted $C_{66}$ as relative change $\mathrm{\Delta }C/C$. The data are shifted for clarity. The vertical arrows indicate phase transitions.

Figure 5

(a) Temperature and (b), (c) field dependences of the transverse elastic modulus $C_{44}$ for $\mathit{H}\left|\right|\left[100\right]$. The curves are vertically offset for clarity. The vertical arrows indicate phase transitions.

Figure 6

(a), (b), (f) Temperature and (c) ,(d), (e) field dependences of the transverse elastic moduli $C_{44}$ and $C_{66}$ for $\mathit{H}\left|\right|\left[001\right]$. The curves are shifted along the $y$ axis for clarity. (d) represents data at $T<15$ K in an expanded scale below 2 T. The data at 9 K for $C_{66}$ are shown for up and down sweeps. The vertical arrows indicate phase transitions.

Figure 7

Temperature dependences of the specific heat for $\mathit{H}\left|\right|\left[001\right]$ (a) from 5 to 25 K and (b) from 15 to 35 K.

Figure 8

$H\text{−}T$ phase diagram for (a) $\mathit{H}\left|\right|\left[100\right]$ and (b) $\mathit{H}\left|\right|\left[001\right]$ of DyNiAl. The dashed lines are guides for the eye. Purple diamonds in (b) represent the phase boundary determined from the specific-heat measurements. Bidirectional blue arrows point out first-order phase transitions. PM and FIP stand for the paramagnetic state and magnetic-field-induced phase, respectively. (c) The same phase diagram for $\mathit{H}\left|\right|\left[001\right]$ in an expanded scale around 15 K and below 2 T.

Figure 9

Temperature dependences of the calculated (a) $C_{44}$ and (b) $C_{66}$ for $\mathit{H}\left|\right|\left[100\right]$, and (c) $C_{44}$ and (d) $C_{66}$ for $\mathit{H}\left|\right|\left[001\right]$.

Figure 10

Field dependences of the calculated CEF energies for (a) $\mathit{H}\left|\right|\left[100\right]$ and (b) $\mathit{H}\left|\right|\left[001\right]$. The two lowest states are depicted by red solid lines. The arrows indicate the field of the level crossing between the two lowest states.