Tunneling and rattling in clathrate crystal

We present tunneling and rattling motions of an off-center guest atom in a cage referring to a clathrate crystal ${\mathrm{La}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$. The elastic constant $C_{44}$ of ${\mathrm{La}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$ shows a Debye-type dispersion of around $20\mathrm{K}$ obeying a relaxation time $\tau ={\tau }_0\exp (E∕k_{\mathrm{B}}T)$ with an attempt time of ${\tau }_0=2.0\times {10}^{-12}\mathrm{s}$ and an activation energy $E=197\mathrm{K}$. At low temperatures below $3\mathrm{K}$ down to $20\mathrm{mK}$, the $C_{44}$ shows a softening of $C_{44}=C_{44}^0(T-T_C^0)∕(T-\Theta )$ with $T_C^0=-337.970\mathrm{mK}$ and $\Theta =-338.044\mathrm{mK}$. These facts are attributed to two different types of off-center motions with ${\Gamma }_5$ symmetry in the $4a$-site cage of ${\mathrm{La}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$, a thermally activated rattling motion over the potential hill, and a tunneling motion through the potential hill at low temperatures.

Figure 1

Temperature dependence of elastic constants $C_{11}$, $(C_{11}-C_{12})∕2$, and $C_{44}$ of ${\mathrm{La}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$. A step-like anomaly in $C_{44}$ around $20\mathrm{K}$ indicates the ultrasonic dispersion.

Figure 2

(a) Frequency dependence of the elastic constant $C_{44}$ measured by the transverse ultrasonic waves with frequencies of $9$, $40$, and $150\mathrm{MHz}$. Arrows in (a) indicate the temperatures satisfying with the resonance condition of $\omega \tau =1$. The inset in (a) represents a fit by the Debye-type dispersion of Eq. (1) for frequencies of $9$, $40$, and $150\mathrm{MHz}$. (b) shows the Arrhenius plot of relaxation time $\tau ={\tau }_0\exp (E∕k_{\mathrm{B}}T)$ in ${\mathrm{La}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$ together with the results of ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$ and $\mathrm{Pr}{\mathrm{Os}}_4{\mathrm{Sb}}_{12}$ (Refs. 9, 11).

Figure 3

The low-temperature softening in elastic constant $C_{44}$ of ${\mathrm{La}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$ below $3\mathrm{K}$ down to $20\mathrm{mK}$. The solid line is a fit by $C_{44}=C_{44}^0(T-T_C^0)∕(T-\Theta )$ with the parameters in the text. Inset presents low-temperature behavior of $C_{44}$ and $(C_{11}-C_{12})∕2$.

Figure 4

(Color) A schematic view of the ${\Gamma }_5$ off-center mode ${\rho }_{\Gamma 5,xy}={\rho }_1+{\rho }_2-{\rho }_3-{\rho }_4+{\rho }_5+{\rho }_6-{\rho }_7-{\rho }_8$ possessing fractional states of the ${\mathrm{La}}_1$ ion over eight off-center potential minima along the threefold [111] axis in a $4a$-site cage of ${\mathrm{La}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$.