Evidence for the Single-Site Quadrupolar Kondo Effect in the Dilute Non-Kramers System ${\mathrm{Y}}_{1-x}{\mathrm{Pr}}_x{\mathrm{Ir}}_2{\mathrm{Zn}}_{20}$

Acoustic signatures of the single-site quadrupolar Kondo effect in ${\mathrm{Y}}_{0.966}{\mathrm{Pr}}_{0.034}{\mathrm{Ir}}_2{\mathrm{Zn}}_{20}$ are presented. The elastic constant $(C_{11}-C_{12})/2$, corresponding to the ${\mathrm{\Gamma }}_3(E)$-symmetry electric-quadrupolar response, reveals a logarithmic temperature dependence of the quadrupolar susceptibility in the low-magnetic-field region below $\sim 0.3\mathrm{K}$. Furthermore, the Curie-type divergence of the elastic constant down to $\sim 1\mathrm{K}$ indicates that the Pr ions in this diluted system have a non-Kramers ground-state doublet. These observations evidence the single-site quadrupolar Kondo effect, as previously suggested based on specific-heat and electrical-resistivity data.

Figure 1

Crystal structure of $\mathrm{Pr}{T}_2{X}_{20}$ ($T=\mathrm{transition}\mathrm{metal}$, $X=\mathrm{Al}$, Zn, and Cd). The Wycoff position and point-group symmetry of the atoms are also represented below each annotation.

Figure 2

Relative change of the elastic constant $(C_{11}-C_{12})/2$, as a function of temperature for various magnetic fields up to 14 T for $H\parallel [001]$ of ${\mathrm{Y}}_{0.966}{\mathrm{Pr}}_{0.034}{\mathrm{Ir}}_2{\mathrm{Zn}}_{20}$. The inset shows $(C_{11}-C_{12})/2$ as a function of magnetic field at various fixed temperatures between 0.04 and 20 K.

Figure 3

Elastic constant $(C_{11}-C_{12})/2$ (open symbols) of ${\mathrm{Y}}_{0.966}{\mathrm{Pr}}_{0.034}{\mathrm{Ir}}_2{\mathrm{Zn}}_{20}$ as a function of temperature at various magnetic fields $H\parallel [001]$. Dashed and solid curves indicate the calculated quadrupolar susceptibility with and without the ${}^{141}\mathrm{Pr}$’s nuclear dipolar contribution, respectively (see text). The insets show enlargements of (a) the calculations below 0.1 K for 0 and 1 T, and (b) the data above 3 T.

Figure 4

(Upper panel) Temperature dependence of the background-subtracted elastic constant $(C_{11}-C_{12})/2$, shown as relative change, at selected magnetic fields. Data of ${\mathrm{PrIr}}_2{\mathrm{Zn}}_{20}$ ($x=1$) [27] and ${\mathrm{YIr}}_2{\mathrm{Zn}}_{20}$ ($x=0$) are also displayed for comparison. The data are vertically shifted to facilitate display of the temperature and magnetic field variations. (Lower panel) Ultrasonic attenuation coefficient vs temperature.