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Register a new userNon-Kramers doublets in PrNi$_2$Cd$_{20}$ and PrPd$_2$Cd$_{20}$
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Praseodymium-based 1-2-20 cage compounds Pr$T_2X_{20}$ ($T$ is generally Ti, V, Nb, Ru, Rh, Ir; and $X$ is either Al, Zn or Cd) provide yet another platform to study non-trivial electronic states of matter ranging from topological and magnetic orders to unconventional multipolar orders and superconductivity. In this paper, we report measurements of the electronic heat capacity in two Pr-based 1-2-20 materials: PrNi$_2$Cd$_{20}$ and PrPd$_2$Cd$_{20}$. We find that the lowest energy multiplet of the Pr $4f^2$ valence configuration is a $Gamma_3$ non-Kramers doublet and can, therefore, be described as a two-level system. By analyzing the dependence of the energy splitting between the ground and first excited singlet states on external magnetic field, we found that the interactions between the two-level systems are weak in PrNi$_2$Cd$_{20}$. However, in PrPd$_2$Cd$_{20}$, the exchange interactions that ultimately promote magnetic or multipolar order are strong enough and must be taken into account to accurately describe the dependence of the energy level splitting on external magnetic field.
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In this study, ultrasonic measurements were performed on a single crystal of cubic PrNi$_2$Cd$_{20}$, down to a temperature of 0.02 K, to investigate the crystalline electric field ground state and search for possible phase transitions at low temperatures. The elastic constant $(C_{11}-C_{12})/2$, which is related to the $Gamma_3$-symmetry quadrupolar response, exhibits the Curie-type softening at temperatures below $sim$30 K, which indicates that the present system has a $Gamma_3$ non-Kramers doublet ground state. A leveling-off of the elastic response appears below $sim$0.1 K toward the lowest temperatures, which implies the presence of level splitting owing to a long-range order in a finite-volume fraction associated with $Gamma_3$-symmetry multipoles. A magnetic field-temperature phase diagram of the present compound is constructed up to 28 T for $H parallel$ [110]. A clear acoustic de Haas-van Alphen signal and a possible magnetic-field-induced phase transition at $H sim$26 T are also detected by high-magnetic-field measurements.
Ultrasonic investigations of the single-site quadrupolar Kondo effect in diluted Pr system Y$_{0.966}$Pr$_{0.034}$Ir$_2$Zn$_{20}$ are reported. The elastic constant $(C_{11}-C_{12})/2$ is measured down to ~40 mK using ultrasound for the dilute system Y$_{0.966}$Pr$_{0.034}$Ir$_2$Zn$_{20}$ and the pure compound YIr$_2$Zn$_{20}$. We found that the elastic constant $(C_{11}-C_{12})/2$ of the Pr-dilute system exhibits a logarithmic temperature dependence below $T_0$ ~0.3 K, where non-Fermi-liquid (NFL) behavior in the specific heat and electrical resistivity is observed. This logarithmic temperature variation manifested in the $Gamma_3$-symmetry quadrupolar susceptibility is consistent with the theoretical prediction of the quadrupolar Kondo effect by D. L. Cox. On the other hand, the pure compound YIr$_2$Zn$_{20}$ without $4f$-electron contributions shows nearly no change in its elastic constants evidencing negligible phonon contributions. In addition, clear acoustic de Haas-van Alphen (dHvA) oscillations in the elastic constant were detected for both compounds on applying magnetic field. This is mainly interpreted as contribution from the Fermi surface of YIr$_2$Zn$_{20}$.
The authors reply to the Comment arXiv:2104.03770 by P. Canfield et. al.
We report thermal expansion and magnetostriction of the cubic non-Kramers system PrIr$_2$Zn$_{20}$ with a non-magnetic $varGamma_{3}$ ground state doublet. In previous experiments, antiferroquadrupolar order at hbox{$T_{mathrm{Q}}=0.11$,K} and a Fermi liquid state around $B_{mathrm{c}}approx5$,T for hbox{$boldsymbol{B}parallel[001]$}, indicative of possible ferrohastatic order, were discovered. For magnetic fields hbox{$boldsymbol{B}parallel[001]$}, the low temperature longitudinal and transverse thermal expansion and magnetostriction are highly anisotropic. The resulting volume strain is very small, indicating that the Pr valence remains nearly constant as a function of magnetic field. We conclude that the Fermi liquid state around $B_{mathrm{c}}$ forms through a very little change in c-f hybridization. This result is in sharp contrast to Ce- and Yb-based Kramers Kondo lattices which show significantly larger volume strains due to the high sensitivity of the Kondo temperature to hydrostatic pressure.
Acoustic signatures of the single-site quadrupolar Kondo effect in Y$_{0.966}$Pr$_{0.034}$Ir$_2$Zn$_{20}$ are presented. The elastic constant ($C_{11}-C_{12}$)/2, corresponding to the $Gamma_3$(E)-symmetry electric-quadrupolar response, reveals a logarithmic temperature dependence of the quadrupolar susceptibility in the low-magnetic-field region below $sim$0.3 K. Furthermore, the Curie-type divergence of the elastic constant down to $sim$1 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.
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