Local-symmetry-sensitive elastic softening in the Kramers doublet system ${\mathrm{Y}}_{1-x}{\mathrm{Nd}}_x{\mathrm{Co}}_2{\mathrm{Zn}}_{20}$

We investigated the elastic properties of ${\mathrm{Y}}_{1-x}{\mathrm{Nd}}_x{\mathrm{Co}}_2{\mathrm{Zn}}_{20}$ with localized Nd $f$ electrons and ground-state Kramers doublet. All longitudinal and transverse moduli of ${\mathrm{NdCo}}_2{\mathrm{Zn}}_{20}$ ($x=$ 1) show an elastic softening below 50 K accompanied by a minimum around 2.5 K. The softening, which is robust to magnetic fields up to 8 T, is not observed for samples with Nd concentrations of $x=0.19$, 0.05, and 0. In localized $f$ electron systems, elastic softening from high temperatures is often understood by crystal electric field effects; however, this cannot explain the behavior in ${\mathrm{NdCo}}_2{\mathrm{Zn}}_{20}$. Our experimental and calculated results reveal that the softening neither is caused by a phonon contribution, a ${\mathrm{Nd}}^{3+}$ single-site effect, nor a magnetic interaction. We conclude that the softening is due to a local-symmetry-sensitive electronic state in ${\mathrm{NdCo}}_2{\mathrm{Zn}}_{20}$.

Figure 1

Temperature dependence of the longitudinal elastic modulus $C_{11}$ (upper panel) as well as the transverse moduli $C_{44}$ (middle panel) and ($C_{11}-C_{12}$)/2 (lower panel) of ${\mathrm{NdCo}}_2{\mathrm{Zn}}_{20}$ ($x=$ 1). The insets display the same data on an expanded temperature scale below 5 K. The red solid and blue dashed lines represent the fit result and the background stiffness, respectively, calculated using the CEF model as discussed in the text. (Notice: from our careful new ultrasonic measurements, we corrected the absolute value of the sound velocity in $C_{44}$ from the data previously reported [22].)

Figure 2

Temperature dependence of the longitudinal elastic modulus $C_{11}$ (upper panel) as well as the transverse moduli $C_{44}$ (middle panel) and ($C_{11}-C_{12}$)/2 (lower panel) in magnetic fields of 4 and 8 T applied along [001] in ${\mathrm{NdCo}}_2{\mathrm{Zn}}_{20}$ ($x=$ 1). We plotted the moduli as relative change $\mathrm{\Delta }C/C$. The data in $H=$ 4 and 8 T are vertically offset for clarity. The insets represent the data on an expanded temperature scale below 5 K. The right inset in the lower panel shows the bulk modulus $C_{\mathrm{B}}$ determined for zero magnetic field.

Figure 3

Temperature dependence of the longitudinal elastic modulus $C_{11}$ (upper panel) as well as the transverse moduli $C_{44}$ (middle panel) and ($C_{11}-C_{12}$)/2 (lower panel) for ${\mathrm{Y}}_{1-x}{\mathrm{Nd}}_x{\mathrm{Co}}_2{\mathrm{Zn}}_{20}$ ($x=1$, 0.19, 0.05, and 0). The data are shifted along the vertical axis for clarity.