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Pressure induced quenching of planar rattling in Cu$_{10}$Zn$_{2}$Sb$_{4}$S$_{13}$ studied by specific-heat and x-ray diffraction measurements

We have studied the pressure effect on the rattling of tetrahedrite Cu$_{10}$Zn$_{2}$Sb$_{4}$S$_{13\,}$(CZSS) and type-I clathrate Ba$_{8}$Ga$_{16}$Sn$_{30\,}$(BGS) by specific heat and x-ray diffraction measurements. By applying pressure $P$, the rattling energy for CZSS initially decreases and steeply increases for $P$ $\textgreater$ $1$ GPa. By contrast, the energy for BGS increases monotonically with $P$ up to 6.5 GPa. An analysis of the pressure dependent specific heat and x-ray diffraction indicates that the out-of-plane rattling of the Cu atoms in the S$_{3}$ triangle of CZSS originates from the chemical pressure, unlike the rattling of the Ba ions among off-center sites in an oversized cage of BGS. The rattling in CZSS ceases upon further increasing $P$ above 2 GPa, suggesting that Cu atoms escape away from the S$_{3}$ triangle plane.