Magnetic transition due to the inter-singlet spin-exchange interaction and elastic softening by the interplay of electric quadrupoles in the distorted kagome lattice antiferromagnet ${\mathrm{Tb}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$

The distorted kagome lattice antiferromagnet ${\mathrm{Tb}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$ with a hexagonal structure has the Néel temperature $T_{\mathrm{N}}=22$ K. To clarify the $4f$-electronic state and an influence of electric quadrupoles in ${\mathrm{Tb}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$, ultrasonic measurements on a single-crystalline sample at zero magnetic field and under fields were carried. A characteristic elastic softening of the transverse modulus $C_{66}$ originating from a quadrupole interaction was found. The crystal electric field parameters were determined to reproduce $C_{66}$, magnetic susceptibilities, and magnetization curves. The obtained level scheme is that the ground and first excited states are singlets, despite the existence of both the magnetic transition and the quadrupole interaction, indicating that ${\mathrm{Tb}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$ is a curious compound. The positive sign of the quadrupole-quadrupole coupling constant for $C_{66}$ indicates a ferroquadrupolar-type interaction of the electric quadrupole $O_{xy}$ or $O_2^2$. The anisotropic magnetic field dependencies of $T_{\mathrm{N}}$ in the field along [100] and [001] were also clarified.

Figure 1

$T$ dependencies of the magnetic susceptibility $\chi$ along (a) [100] and (b) [001] on field cooling at elevated magnetic fields. The curves along [001] above 1 T are vertically offset for easier viewing of each data curve. The arrow indicates the phase transition at $T_{\mathrm{N}}$.

Figure 2

$T$ dependencies of the longitudinal elastic moduli $C_{11}$ and $C_{33}$ in ${\mathrm{Tb}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$. The vertical arrow indicates the phase transition at $T_{\mathrm{N}}$.

Figure 3

$T$ dependencies of the transverse elastic moduli $C_{44}$ and $C_{66}$ in ${\mathrm{Tb}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$. The vertical arrow indicates the phase transition at $T_{\mathrm{N}}$. The red solid and blue broken curves represent the fitting result and the background stiffness, respectively.

Figure 4

(a) $T$ dependencies of the inverse magnetic susceptibility at 0.1 T. (b) Magnetization curves at 2 K. The calculated results are represented by solid curves. The data of $M$ are the same with the data previously reported [12]. (c) The $4f$-level scheme of ${\mathrm{Tb}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$ obtained from the CEF parameters listed in Table 1.

Figure 5

$T$ and $H$ dependencies of the elastic moduli (a) and (b) $C_{66}$ and (c) $C_{11}$ along $H\left|\right|\left[100\right]$ at various conditions in ${\mathrm{Tb}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$. The data are plotted adding a constant value to easily see each data curve. The arrow indicates the phase transition at $T_{\mathrm{N}}$. (d) The $H-T$ phase diagram along $H\left|\right|\left[100\right]$. The broken curve is a guide for the eyes. Black circles and magenta down triangles (red squares and blue up triangles) represent the phase boundary determined by the $T$ ($H$) dependencies of the elastic moduli. NOS indicates the nonordered state.

Figure 6

$T$ and $H$ dependencies of the elastic moduli (a) and (b) $C_{33}$ and (c) $C_{44}$ along $H\left|\right|\left[001\right]$ at various conditions in ${\mathrm{Tb}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$. We plotted $\mathrm{\Delta }C/C$ by adding a constant value to easily see each data curve. The arrows indicate the phase transitions. (d) The $H-T$ phase diagram along $H\left|\right|\left[001\right]$. The broken curves are guides for the eyes. The middle point of hysteresis is defined as the phase transition, and hysteresis is depicted by an error bar. Black circles and magenta down triangles (red squares and green up triangles) represent the phase boundary determined by the $T$ ($H$) dependencies of the elastic moduli.

Figure 7

The $H$ dependencies of the calculated CEF energies along (a) $H\left|\right|\left[100\right]$ and (b) $H\left|\right|\left[001\right]$. The ground singlet is depicted by the red solid curve.