Using density-functional theory we compute the formation energies of vacancies $\square$ in the framework of type I ${\mathrm{Ba}}_8{\mathrm{Zn}}_x{\mathrm{Ge}}_{46-x-y}{\square }_y$ clathrates as a function of Zn substitutions. While resolving the contradiction on the experimental reports on the relation between the concentration of Zn, $x$, and the number of framework vacancies, $y$, our study confirms the observations that vacancies are destabilized with the increase in $x$: Up to three vacancies per formula unit can be stabilized for $0\le x<6$, while they are unstable for $x\ge 6$. This behavior arises from the lower formation energy of (charged) Zn substitution in the Ge framework as compared to a (charged) vacancy creation. Finally our study reveals the stability of a high-temperature $T$ phase ${\mathrm{Ba}}_8{\mathrm{Zn}}_6{\mathrm{Ge}}_{40}$ that has high electronic carrier concentration, whereby the carriers also show high-$T$-induced variation effective for its thermoelectric application.

• Received 6 May 2016
• Revised 28 August 2016

DOI:https://doi.org/10.1103/PhysRevB.94.094305